34:I[651451,["/_next/static/chunks/18-nz0ve3_rc0.js?dpl=nogit-1792125771","/_next/static/chunks/2_6olli-sutu3.js?dpl=nogit-1792125771","/_next/static/chunks/0z347qy8pf1j3.js?dpl=nogit-1792125771","/_next/static/chunks/02eggfg7vwd9q.js?dpl=nogit-1792125771","/_next/static/chunks/1gm-hv-3-fp7_.js?dpl=nogit-1792125771","/_next/static/chunks/2xa0kylife8l8.js?dpl=nogit-1792125771","/_next/static/chunks/2lsi6eli88uwt.js?dpl=nogit-1792125771","/_next/static/chunks/0zkcxwtbsqliz.js?dpl=nogit-1792125771","/_next/static/chunks/0kyh6u8hfv06h.js?dpl=nogit-1792125771","/_next/static/chunks/1zirpvj720iz0.js?dpl=nogit-1792125771","/_next/static/chunks/0mt9yp27gb-6w.js?dpl=nogit-1792125771","/_next/static/chunks/38w1x0vrld4-w.js?dpl=nogit-1792125771","/_next/static/chunks/0myxz4gywm9of.js?dpl=nogit-1792125771","/_next/static/chunks/40lyu17d590zu.js?dpl=nogit-1792125771","/_next/static/chunks/1jo7uanvci4-r.js?dpl=nogit-1792125771","/_next/static/chunks/0papbjd-vx5q5.js?dpl=nogit-1792125771","/_next/static/chunks/3ml3vooxhiupb.js?dpl=nogit-1792125771","/_next/static/chunks/11kq2otno6-oj.js?dpl=nogit-1792125771","/_next/static/chunks/23f2tb1apskc8.js?dpl=nogit-1792125771","/_next/static/chunks/04zr-wqafvwoa.js?dpl=nogit-1792125771","/_next/static/chunks/3x5t_ry1ziv7r.js?dpl=nogit-1792125771","/_next/static/chunks/2mv9l1xl_aa4j.js?dpl=nogit-1792125771","/_next/static/chunks/0iapw94rt37n4.js?dpl=nogit-1792125771","/_next/static/chunks/2rvalcou1gw7v.js?dpl=nogit-1792125771","/_next/static/chunks/1c-rbkxmrcq7-.js?dpl=nogit-1792125771","/_next/static/chunks/0v4byuex1x38h.js?dpl=nogit-1792125771","/_next/static/chunks/04wg1hnm8knmc.js?dpl=nogit-1792125771","/_next/static/chunks/3_six85qevnm3.js?dpl=nogit-1792125771","/_next/static/chunks/2dx-vzwyw9a3g.js?dpl=nogit-1792125771","/_next/static/chunks/1ku-vdy4ggo41.js?dpl=nogit-1792125771","/_next/static/chunks/1vovlfobo0rf7.js?dpl=nogit-1792125771","/_next/static/chunks/0llqhg6-bxn_b.js?dpl=nogit-1792125771","/_next/static/chunks/1coblm1-go81_.js?dpl=nogit-1792125771","/_next/static/chunks/1jxfzuqocprar.js?dpl=nogit-1792125771","/_next/static/chunks/0k-y_4mcfub-p.js?dpl=nogit-1792125771","/_next/static/chunks/3bplvoggfmxkd.js?dpl=nogit-1792125771","/_next/static/chunks/1drwxxqzvnao9.js?dpl=nogit-1792125771","/_next/static/chunks/0ltoscgl22itf.js?dpl=nogit-1792125771","/_next/static/chunks/36ldfo6wk-834.js?dpl=nogit-1792125771","/_next/static/chunks/24srd-xd1m5z8.js?dpl=nogit-1792125771","/_next/static/chunks/1zo1304bjzjl_.js?dpl=nogit-1792125771","/_next/static/chunks/13gj59kzq50t_.js?dpl=nogit-1792125771","/_next/static/chunks/0beaxlbowx15e.js?dpl=nogit-1792125771","/_next/static/chunks/3q5gc8lbwr4b4.js?dpl=nogit-1792125771","/_next/static/chunks/2mbz851h6l6ul.js?dpl=nogit-1792125771","/_next/static/chunks/2yss1g0cimuwe.js?dpl=nogit-1792125771","/_next/static/chunks/0ioq6d8hla7h0.js?dpl=nogit-1792125771","/_next/static/chunks/1w193fj-lcfgs.js?dpl=nogit-1792125771","/_next/static/chunks/421rwkxv431i9.js?dpl=nogit-1792125771","/_next/static/chunks/1lsxkfxy262n-.js?dpl=nogit-1792125771","/_next/static/chunks/0zg1om_k0m5zv.js?dpl=nogit-1792125771","/_next/static/chunks/2l63yczi1z2qz.js?dpl=nogit-1792125771","/_next/static/chunks/1d4d88jx43_iw.js?dpl=nogit-1792125771","/_next/static/chunks/33y-ad8441jub.js?dpl=nogit-1792125771","/_next/static/chunks/36nndwkz7iik3.js?dpl=nogit-1792125771","/_next/static/chunks/3xskuweujaw05.js?dpl=nogit-1792125771","/_next/static/chunks/193n44ayumkve.js?dpl=nogit-1792125771","/_next/static/chunks/1krfs1hu3nlgy.js?dpl=nogit-1792125771","/_next/static/chunks/1iyn56mfdoj9b.js?dpl=nogit-1792125771","/_next/static/chunks/1ksyn9m68446d.js?dpl=nogit-1792125771","/_next/static/chunks/3n19bhgn4l5zi.js?dpl=nogit-1792125771","/_next/static/chunks/3e1v2g1f52jrg.js?dpl=nogit-1792125771"],"UserProgressProvider"] 35:I[196793,["/_next/static/chunks/18-nz0ve3_rc0.js?dpl=nogit-1792125771","/_next/static/chunks/2_6olli-sutu3.js?dpl=nogit-1792125771","/_next/static/chunks/0z347qy8pf1j3.js?dpl=nogit-1792125771","/_next/static/chunks/02eggfg7vwd9q.js?dpl=nogit-1792125771","/_next/static/chunks/1gm-hv-3-fp7_.js?dpl=nogit-1792125771","/_next/static/chunks/2xa0kylife8l8.js?dpl=nogit-1792125771","/_next/static/chunks/2lsi6eli88uwt.js?dpl=nogit-1792125771","/_next/static/chunks/0zkcxwtbsqliz.js?dpl=nogit-1792125771","/_next/static/chunks/0kyh6u8hfv06h.js?dpl=nogit-1792125771","/_next/static/chunks/1zirpvj720iz0.js?dpl=nogit-1792125771","/_next/static/chunks/0mt9yp27gb-6w.js?dpl=nogit-1792125771","/_next/static/chunks/38w1x0vrld4-w.js?dpl=nogit-1792125771","/_next/static/chunks/0myxz4gywm9of.js?dpl=nogit-1792125771","/_next/static/chunks/40lyu17d590zu.js?dpl=nogit-1792125771","/_next/static/chunks/1jo7uanvci4-r.js?dpl=nogit-1792125771","/_next/static/chunks/0papbjd-vx5q5.js?dpl=nogit-1792125771","/_next/static/chunks/3ml3vooxhiupb.js?dpl=nogit-1792125771","/_next/static/chunks/11kq2otno6-oj.js?dpl=nogit-1792125771","/_next/static/chunks/23f2tb1apskc8.js?dpl=nogit-1792125771","/_next/static/chunks/04zr-wqafvwoa.js?dpl=nogit-1792125771","/_next/static/chunks/3x5t_ry1ziv7r.js?dpl=nogit-1792125771","/_next/static/chunks/2mv9l1xl_aa4j.js?dpl=nogit-1792125771","/_next/static/chunks/0iapw94rt37n4.js?dpl=nogit-1792125771","/_next/static/chunks/2rvalcou1gw7v.js?dpl=nogit-1792125771","/_next/static/chunks/1c-rbkxmrcq7-.js?dpl=nogit-1792125771","/_next/static/chunks/0v4byuex1x38h.js?dpl=nogit-1792125771","/_next/static/chunks/04wg1hnm8knmc.js?dpl=nogit-1792125771","/_next/static/chunks/3_six85qevnm3.js?dpl=nogit-1792125771","/_next/static/chunks/2dx-vzwyw9a3g.js?dpl=nogit-1792125771","/_next/static/chunks/1ku-vdy4ggo41.js?dpl=nogit-1792125771","/_next/static/chunks/1vovlfobo0rf7.js?dpl=nogit-1792125771","/_next/static/chunks/0llqhg6-bxn_b.js?dpl=nogit-1792125771","/_next/static/chunks/1coblm1-go81_.js?dpl=nogit-1792125771","/_next/static/chunks/1jxfzuqocprar.js?dpl=nogit-1792125771","/_next/static/chunks/0k-y_4mcfub-p.js?dpl=nogit-1792125771","/_next/static/chunks/3bplvoggfmxkd.js?dpl=nogit-1792125771","/_next/static/chunks/1drwxxqzvnao9.js?dpl=nogit-1792125771","/_next/static/chunks/0ltoscgl22itf.js?dpl=nogit-1792125771","/_next/static/chunks/36ldfo6wk-834.js?dpl=nogit-1792125771","/_next/static/chunks/24srd-xd1m5z8.js?dpl=nogit-1792125771","/_next/static/chunks/1zo1304bjzjl_.js?dpl=nogit-1792125771","/_next/static/chunks/13gj59kzq50t_.js?dpl=nogit-1792125771","/_next/static/chunks/0beaxlbowx15e.js?dpl=nogit-1792125771","/_next/static/chunks/3q5gc8lbwr4b4.js?dpl=nogit-1792125771","/_next/static/chunks/2mbz851h6l6ul.js?dpl=nogit-1792125771","/_next/static/chunks/2yss1g0cimuwe.js?dpl=nogit-1792125771","/_next/static/chunks/0ioq6d8hla7h0.js?dpl=nogit-1792125771","/_next/static/chunks/1w193fj-lcfgs.js?dpl=nogit-1792125771","/_next/static/chunks/421rwkxv431i9.js?dpl=nogit-1792125771","/_next/static/chunks/1lsxkfxy262n-.js?dpl=nogit-1792125771","/_next/static/chunks/0zg1om_k0m5zv.js?dpl=nogit-1792125771","/_next/static/chunks/2l63yczi1z2qz.js?dpl=nogit-1792125771","/_next/static/chunks/1d4d88jx43_iw.js?dpl=nogit-1792125771","/_next/static/chunks/33y-ad8441jub.js?dpl=nogit-1792125771","/_next/static/chunks/36nndwkz7iik3.js?dpl=nogit-1792125771","/_next/static/chunks/3xskuweujaw05.js?dpl=nogit-1792125771","/_next/static/chunks/193n44ayumkve.js?dpl=nogit-1792125771","/_next/static/chunks/1krfs1hu3nlgy.js?dpl=nogit-1792125771","/_next/static/chunks/1iyn56mfdoj9b.js?dpl=nogit-1792125771","/_next/static/chunks/1ksyn9m68446d.js?dpl=nogit-1792125771","/_next/static/chunks/3n19bhgn4l5zi.js?dpl=nogit-1792125771","/_next/static/chunks/3e1v2g1f52jrg.js?dpl=nogit-1792125771"],"Gingerbread"] 37:I[716297,["/_next/static/chunks/18-nz0ve3_rc0.js?dpl=nogit-1792125771","/_next/static/chunks/2_6olli-sutu3.js?dpl=nogit-1792125771","/_next/static/chunks/0z347qy8pf1j3.js?dpl=nogit-1792125771","/_next/static/chunks/02eggfg7vwd9q.js?dpl=nogit-1792125771","/_next/static/chunks/1gm-hv-3-fp7_.js?dpl=nogit-1792125771","/_next/static/chunks/2xa0kylife8l8.js?dpl=nogit-1792125771","/_next/static/chunks/2lsi6eli88uwt.js?dpl=nogit-1792125771","/_next/static/chunks/0zkcxwtbsqliz.js?dpl=nogit-1792125771","/_next/static/chunks/0kyh6u8hfv06h.js?dpl=nogit-1792125771","/_next/static/chunks/1zirpvj720iz0.js?dpl=nogit-1792125771","/_next/static/chunks/0mt9yp27gb-6w.js?dpl=nogit-1792125771","/_next/static/chunks/38w1x0vrld4-w.js?dpl=nogit-1792125771","/_next/static/chunks/0myxz4gywm9of.js?dpl=nogit-1792125771","/_next/static/chunks/40lyu17d590zu.js?dpl=nogit-1792125771","/_next/static/chunks/1jo7uanvci4-r.js?dpl=nogit-1792125771","/_next/static/chunks/0papbjd-vx5q5.js?dpl=nogit-1792125771","/_next/static/chunks/3ml3vooxhiupb.js?dpl=nogit-1792125771","/_next/static/chunks/11kq2otno6-oj.js?dpl=nogit-1792125771","/_next/static/chunks/23f2tb1apskc8.js?dpl=nogit-1792125771","/_next/static/chunks/04zr-wqafvwoa.js?dpl=nogit-1792125771","/_next/static/chunks/3x5t_ry1ziv7r.js?dpl=nogit-1792125771","/_next/static/chunks/2mv9l1xl_aa4j.js?dpl=nogit-1792125771","/_next/static/chunks/0iapw94rt37n4.js?dpl=nogit-1792125771","/_next/static/chunks/2rvalcou1gw7v.js?dpl=nogit-1792125771","/_next/static/chunks/1c-rbkxmrcq7-.js?dpl=nogit-1792125771","/_next/static/chunks/0v4byuex1x38h.js?dpl=nogit-1792125771","/_next/static/chunks/04wg1hnm8knmc.js?dpl=nogit-1792125771","/_next/static/chunks/3_six85qevnm3.js?dpl=nogit-1792125771","/_next/static/chunks/2dx-vzwyw9a3g.js?dpl=nogit-1792125771","/_next/static/chunks/1ku-vdy4ggo41.js?dpl=nogit-1792125771","/_next/static/chunks/1vovlfobo0rf7.js?dpl=nogit-1792125771","/_next/static/chunks/0llqhg6-bxn_b.js?dpl=nogit-1792125771","/_next/static/chunks/1coblm1-go81_.js?dpl=nogit-1792125771","/_next/static/chunks/1jxfzuqocprar.js?dpl=nogit-1792125771","/_next/static/chunks/0k-y_4mcfub-p.js?dpl=nogit-1792125771","/_next/static/chunks/3bplvoggfmxkd.js?dpl=nogit-1792125771","/_next/static/chunks/1drwxxqzvnao9.js?dpl=nogit-1792125771","/_next/static/chunks/0ltoscgl22itf.js?dpl=nogit-1792125771","/_next/static/chunks/36ldfo6wk-834.js?dpl=nogit-1792125771","/_next/static/chunks/24srd-xd1m5z8.js?dpl=nogit-1792125771","/_next/static/chunks/1zo1304bjzjl_.js?dpl=nogit-1792125771","/_next/static/chunks/13gj59kzq50t_.js?dpl=nogit-1792125771","/_next/static/chunks/0beaxlbowx15e.js?dpl=nogit-1792125771","/_next/static/chunks/3q5gc8lbwr4b4.js?dpl=nogit-1792125771","/_next/static/chunks/2mbz851h6l6ul.js?dpl=nogit-1792125771","/_next/static/chunks/2yss1g0cimuwe.js?dpl=nogit-1792125771","/_next/static/chunks/0ioq6d8hla7h0.js?dpl=nogit-1792125771","/_next/static/chunks/1w193fj-lcfgs.js?dpl=nogit-1792125771","/_next/static/chunks/421rwkxv431i9.js?dpl=nogit-1792125771","/_next/static/chunks/1lsxkfxy262n-.js?dpl=nogit-1792125771","/_next/static/chunks/0zg1om_k0m5zv.js?dpl=nogit-1792125771","/_next/static/chunks/2l63yczi1z2qz.js?dpl=nogit-1792125771","/_next/static/chunks/1d4d88jx43_iw.js?dpl=nogit-1792125771","/_next/static/chunks/33y-ad8441jub.js?dpl=nogit-1792125771","/_next/static/chunks/36nndwkz7iik3.js?dpl=nogit-1792125771","/_next/static/chunks/3xskuweujaw05.js?dpl=nogit-1792125771","/_next/static/chunks/193n44ayumkve.js?dpl=nogit-1792125771","/_next/static/chunks/1krfs1hu3nlgy.js?dpl=nogit-1792125771","/_next/static/chunks/1iyn56mfdoj9b.js?dpl=nogit-1792125771","/_next/static/chunks/1ksyn9m68446d.js?dpl=nogit-1792125771","/_next/static/chunks/3n19bhgn4l5zi.js?dpl=nogit-1792125771","/_next/static/chunks/3e1v2g1f52jrg.js?dpl=nogit-1792125771"],"TopicRecentlyViewed"] 38:I[647935,["/_next/static/chunks/18-nz0ve3_rc0.js?dpl=nogit-1792125771","/_next/static/chunks/2_6olli-sutu3.js?dpl=nogit-1792125771","/_next/static/chunks/0z347qy8pf1j3.js?dpl=nogit-1792125771","/_next/static/chunks/02eggfg7vwd9q.js?dpl=nogit-1792125771","/_next/static/chunks/1gm-hv-3-fp7_.js?dpl=nogit-1792125771","/_next/static/chunks/2xa0kylife8l8.js?dpl=nogit-1792125771","/_next/static/chunks/2lsi6eli88uwt.js?dpl=nogit-1792125771","/_next/static/chunks/0zkcxwtbsqliz.js?dpl=nogit-1792125771","/_next/static/chunks/0kyh6u8hfv06h.js?dpl=nogit-1792125771","/_next/static/chunks/1zirpvj720iz0.js?dpl=nogit-1792125771","/_next/static/chunks/0mt9yp27gb-6w.js?dpl=nogit-1792125771","/_next/static/chunks/38w1x0vrld4-w.js?dpl=nogit-1792125771","/_next/static/chunks/0myxz4gywm9of.js?dpl=nogit-1792125771","/_next/static/chunks/40lyu17d590zu.js?dpl=nogit-1792125771","/_next/static/chunks/1jo7uanvci4-r.js?dpl=nogit-1792125771","/_next/static/chunks/0papbjd-vx5q5.js?dpl=nogit-1792125771","/_next/static/chunks/3ml3vooxhiupb.js?dpl=nogit-1792125771","/_next/static/chunks/11kq2otno6-oj.js?dpl=nogit-1792125771","/_next/static/chunks/23f2tb1apskc8.js?dpl=nogit-1792125771","/_next/static/chunks/04zr-wqafvwoa.js?dpl=nogit-1792125771","/_next/static/chunks/3x5t_ry1ziv7r.js?dpl=nogit-1792125771","/_next/static/chunks/2mv9l1xl_aa4j.js?dpl=nogit-1792125771","/_next/static/chunks/0iapw94rt37n4.js?dpl=nogit-1792125771","/_next/static/chunks/2rvalcou1gw7v.js?dpl=nogit-1792125771","/_next/static/chunks/1c-rbkxmrcq7-.js?dpl=nogit-1792125771","/_next/static/chunks/0v4byuex1x38h.js?dpl=nogit-1792125771","/_next/static/chunks/04wg1hnm8knmc.js?dpl=nogit-1792125771","/_next/static/chunks/3_six85qevnm3.js?dpl=nogit-1792125771","/_next/static/chunks/2dx-vzwyw9a3g.js?dpl=nogit-1792125771","/_next/static/chunks/1ku-vdy4ggo41.js?dpl=nogit-1792125771","/_next/static/chunks/1vovlfobo0rf7.js?dpl=nogit-1792125771","/_next/static/chunks/0llqhg6-bxn_b.js?dpl=nogit-1792125771","/_next/static/chunks/1coblm1-go81_.js?dpl=nogit-1792125771","/_next/static/chunks/1jxfzuqocprar.js?dpl=nogit-1792125771","/_next/static/chunks/0k-y_4mcfub-p.js?dpl=nogit-1792125771","/_next/static/chunks/3bplvoggfmxkd.js?dpl=nogit-1792125771","/_next/static/chunks/1drwxxqzvnao9.js?dpl=nogit-1792125771","/_next/static/chunks/0ltoscgl22itf.js?dpl=nogit-1792125771","/_next/static/chunks/36ldfo6wk-834.js?dpl=nogit-1792125771","/_next/static/chunks/24srd-xd1m5z8.js?dpl=nogit-1792125771","/_next/static/chunks/1zo1304bjzjl_.js?dpl=nogit-1792125771","/_next/static/chunks/13gj59kzq50t_.js?dpl=nogit-1792125771","/_next/static/chunks/0beaxlbowx15e.js?dpl=nogit-1792125771","/_next/static/chunks/3q5gc8lbwr4b4.js?dpl=nogit-1792125771","/_next/static/chunks/2mbz851h6l6ul.js?dpl=nogit-1792125771","/_next/static/chunks/2yss1g0cimuwe.js?dpl=nogit-1792125771","/_next/static/chunks/0ioq6d8hla7h0.js?dpl=nogit-1792125771","/_next/static/chunks/1w193fj-lcfgs.js?dpl=nogit-1792125771","/_next/static/chunks/421rwkxv431i9.js?dpl=nogit-1792125771","/_next/static/chunks/1lsxkfxy262n-.js?dpl=nogit-1792125771","/_next/static/chunks/0zg1om_k0m5zv.js?dpl=nogit-1792125771","/_next/static/chunks/2l63yczi1z2qz.js?dpl=nogit-1792125771","/_next/static/chunks/1d4d88jx43_iw.js?dpl=nogit-1792125771","/_next/static/chunks/33y-ad8441jub.js?dpl=nogit-1792125771","/_next/static/chunks/36nndwkz7iik3.js?dpl=nogit-1792125771","/_next/static/chunks/3xskuweujaw05.js?dpl=nogit-1792125771","/_next/static/chunks/193n44ayumkve.js?dpl=nogit-1792125771","/_next/static/chunks/1krfs1hu3nlgy.js?dpl=nogit-1792125771","/_next/static/chunks/1iyn56mfdoj9b.js?dpl=nogit-1792125771","/_next/static/chunks/1ksyn9m68446d.js?dpl=nogit-1792125771","/_next/static/chunks/3n19bhgn4l5zi.js?dpl=nogit-1792125771","/_next/static/chunks/3e1v2g1f52jrg.js?dpl=nogit-1792125771"],"RecordResourceResume"] 39:I[426249,["/_next/static/chunks/18-nz0ve3_rc0.js?dpl=nogit-1792125771","/_next/static/chunks/2_6olli-sutu3.js?dpl=nogit-1792125771","/_next/static/chunks/0z347qy8pf1j3.js?dpl=nogit-1792125771","/_next/static/chunks/02eggfg7vwd9q.js?dpl=nogit-1792125771","/_next/static/chunks/1gm-hv-3-fp7_.js?dpl=nogit-1792125771","/_next/static/chunks/2xa0kylife8l8.js?dpl=nogit-1792125771","/_next/static/chunks/2lsi6eli88uwt.js?dpl=nogit-1792125771","/_next/static/chunks/0zkcxwtbsqliz.js?dpl=nogit-1792125771","/_next/static/chunks/0kyh6u8hfv06h.js?dpl=nogit-1792125771","/_next/static/chunks/1zirpvj720iz0.js?dpl=nogit-1792125771","/_next/static/chunks/0mt9yp27gb-6w.js?dpl=nogit-1792125771","/_next/static/chunks/38w1x0vrld4-w.js?dpl=nogit-1792125771","/_next/static/chunks/0myxz4gywm9of.js?dpl=nogit-1792125771","/_next/static/chunks/40lyu17d590zu.js?dpl=nogit-1792125771","/_next/static/chunks/1jo7uanvci4-r.js?dpl=nogit-1792125771","/_next/static/chunks/0papbjd-vx5q5.js?dpl=nogit-1792125771","/_next/static/chunks/3ml3vooxhiupb.js?dpl=nogit-1792125771","/_next/static/chunks/11kq2otno6-oj.js?dpl=nogit-1792125771","/_next/static/chunks/23f2tb1apskc8.js?dpl=nogit-1792125771","/_next/static/chunks/04zr-wqafvwoa.js?dpl=nogit-1792125771","/_next/static/chunks/3x5t_ry1ziv7r.js?dpl=nogit-1792125771","/_next/static/chunks/2mv9l1xl_aa4j.js?dpl=nogit-1792125771","/_next/static/chunks/0iapw94rt37n4.js?dpl=nogit-1792125771","/_next/static/chunks/2rvalcou1gw7v.js?dpl=nogit-1792125771","/_next/static/chunks/1c-rbkxmrcq7-.js?dpl=nogit-1792125771","/_next/static/chunks/0v4byuex1x38h.js?dpl=nogit-1792125771","/_next/static/chunks/04wg1hnm8knmc.js?dpl=nogit-1792125771","/_next/static/chunks/3_six85qevnm3.js?dpl=nogit-1792125771","/_next/static/chunks/2dx-vzwyw9a3g.js?dpl=nogit-1792125771","/_next/static/chunks/1ku-vdy4ggo41.js?dpl=nogit-1792125771","/_next/static/chunks/1vovlfobo0rf7.js?dpl=nogit-1792125771","/_next/static/chunks/0llqhg6-bxn_b.js?dpl=nogit-1792125771","/_next/static/chunks/1coblm1-go81_.js?dpl=nogit-1792125771","/_next/static/chunks/1jxfzuqocprar.js?dpl=nogit-1792125771","/_next/static/chunks/0k-y_4mcfub-p.js?dpl=nogit-1792125771","/_next/static/chunks/3bplvoggfmxkd.js?dpl=nogit-1792125771","/_next/static/chunks/1drwxxqzvnao9.js?dpl=nogit-1792125771","/_next/static/chunks/0ltoscgl22itf.js?dpl=nogit-1792125771","/_next/static/chunks/36ldfo6wk-834.js?dpl=nogit-1792125771","/_next/static/chunks/24srd-xd1m5z8.js?dpl=nogit-1792125771","/_next/static/chunks/1zo1304bjzjl_.js?dpl=nogit-1792125771","/_next/static/chunks/13gj59kzq50t_.js?dpl=nogit-1792125771","/_next/static/chunks/0beaxlbowx15e.js?dpl=nogit-1792125771","/_next/static/chunks/3q5gc8lbwr4b4.js?dpl=nogit-1792125771","/_next/static/chunks/2mbz851h6l6ul.js?dpl=nogit-1792125771","/_next/static/chunks/2yss1g0cimuwe.js?dpl=nogit-1792125771","/_next/static/chunks/0ioq6d8hla7h0.js?dpl=nogit-1792125771","/_next/static/chunks/1w193fj-lcfgs.js?dpl=nogit-1792125771","/_next/static/chunks/421rwkxv431i9.js?dpl=nogit-1792125771","/_next/static/chunks/1lsxkfxy262n-.js?dpl=nogit-1792125771","/_next/static/chunks/0zg1om_k0m5zv.js?dpl=nogit-1792125771","/_next/static/chunks/2l63yczi1z2qz.js?dpl=nogit-1792125771","/_next/static/chunks/1d4d88jx43_iw.js?dpl=nogit-1792125771","/_next/static/chunks/33y-ad8441jub.js?dpl=nogit-1792125771","/_next/static/chunks/36nndwkz7iik3.js?dpl=nogit-1792125771","/_next/static/chunks/3xskuweujaw05.js?dpl=nogit-1792125771","/_next/static/chunks/193n44ayumkve.js?dpl=nogit-1792125771","/_next/static/chunks/1krfs1hu3nlgy.js?dpl=nogit-1792125771","/_next/static/chunks/1iyn56mfdoj9b.js?dpl=nogit-1792125771","/_next/static/chunks/1ksyn9m68446d.js?dpl=nogit-1792125771","/_next/static/chunks/3n19bhgn4l5zi.js?dpl=nogit-1792125771","/_next/static/chunks/3e1v2g1f52jrg.js?dpl=nogit-1792125771"],"TopicQuestionBank"] 25:["$","$L34",null,{"bundle":{"topicIds":["2af73925-f8b7-4780-a294-32d3d331b72e"],"paperIds":[],"stageIds":[],"stageTopicIds":[],"stageAssessmentIds":[],"topicNoteIds":[]},"children":[["$","$L35",null,{"feature":"question-bank"}],"$L36",["$","$L37",null,{"subjectCode":"a_level_maths__ocr_mei","subjectName":"Maths","topicSlug":"quotient-rule","topicTitle":"1.9.13 Quotient rule (A-level only)","sectionTitle":null,"activity":"worksheet","resourceTitle":"1.9.13 Quotient rule (A-level only) Questions","resourceHref":"/a-level/maths/ocr/mei/quotient-rule/questions/q1","saveTrail":false}],["$","$L38",null,{"href":"/a-level/maths/ocr/mei/quotient-rule/questions/q1","label":"1.9.13 Quotient rule (A-level only)","subjectCode":"a_level_maths__ocr_mei","surface":"questions"}],["$","script",null,{"type":"application/ld+json","dangerouslySetInnerHTML":{"__html":"{\"@context\":\"https://schema.org\",\"@type\":\"LearningResource\",\"name\":\"Question 17 – 1.9.13 Quotient rule (A-level only) - Questions\",\"description\":\"Practise 1.9.13 Quotient rule (A-level only) A Level Maths exam-style questions with worked solutions on MathsGenie.\",\"learningResourceType\":\"Practice questions\",\"educationalLevel\":\"A Level\",\"about\":{\"@type\":\"Thing\",\"name\":\"1.9.13 Quotient rule (A-level only)\"},\"url\":\"https://mathsgenie.co.uk/a-level/maths/ocr/mei/quotient-rule/questions/q17\",\"inLanguage\":\"en-GB\",\"provider\":{\"@id\":\"https://mathsgenie.co.uk/#organization\"}}"}}],["$","div",null,{"className":"min-h-0 w-full flex-1 overflow-hidden bg-background","children":["$","$L39",null,{"backHref":"/a-level/maths/ocr/mei/questions","basePath":"/a-level/maths/ocr/mei/quotient-rule/questions","formulaUrl":null,"initialIndex":16,"initialQuestionWindow":[{"id":"0de699ee-92d8-42c4-a6ff-32b84bb6e495","revisionId":"d65365e0-81a2-451f-8e29-7ed10d973d4c","questionNumber":null,"questionType":"LA","specification":"","difficultyBand":"easy","tier":null,"totalMarks":4,"sourceTitle":null,"calculatorUse":null,"isCalculation":true,"parts":[{"id":"521a3e89-bc6f-472d-af92-1bd34422b69c","label":null,"content":"A robotic arm's torque $\\tau$ (in Newton-metres) during a specific maneuver is modeled as a function of the joint angle $\\theta$ (in radians) by: \\[\\tau(\\theta) = \\frac{4\\theta^3 + 5\\theta}{\\sin \\theta}, \\quad 0 < \\theta < \\pi\\] Find an expression for the rate of change of torque with respect to the angle, $\\frac{d\\tau}{d\\theta}$.","metadata":null,"markscheme":"- Identifies $u = 4\\theta^3 + 5\\theta$ and $v = \\sin \\theta$ M1\n- Differentiates component functions to find $\\frac{du}{d\\theta} = 12\\theta^2 + 5$ and $\\frac{dv}{d\\theta} = \\cos \\theta$ A1\n- Applies the quotient rule formula: $\\frac{d\\tau}{d\\theta} = \\frac{v \\frac{du}{d\\theta} - u \\frac{dv}{d\\theta}}{v^2}$ M1\n- Substitutes correctly to obtain $\\frac{(12\\theta^2 + 5)\\sin \\theta - (4\\theta^3 + 5\\theta)\\cos \\theta}{\\sin^2 \\theta}$ A1","marks":4,"order":0,"markingRubric":null,"tableImages":[],"explanation":{"methods":[{"id":"quotient_rule","label":"Quotient Rule","steps":[{"type":"text","order":0,"title":"Identify quotient components","content":"To find the rate of change of torque, $\\frac{d\\tau}{d\\theta}$, we can apply the **Quotient Rule** because the function $\\tau(\\theta)$ is expressed as one function divided by another:\n\n$$\\tau(\\theta) = \\frac{u}{v} = \\frac{4\\theta^3 + 5\\theta}{\\sin \\theta}$$\n\nFirst, let's identify our numerator, $u$, and our denominator, $v$:\n\n$$u = 4\\theta^3 + 5\\theta$$\n$$v = \\sin \\theta$$\n\n*(This step earns the first method mark **M1**)*","highlights":[{"text":"\\tau(\\theta) = \\frac{4\\theta^3 + 5\\theta}{\\sin \\theta}","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\text{Numerator: } \\htmlData{anchor=u-fn}{u} = 4\\theta^3 + 5\\theta \\implies \\htmlData{anchor=du-deriv}{\\frac{du}{d\\theta}} = 12\\theta^2 + 5$"},{"latex":"$$\\text{Denominator: } \\htmlData{anchor=v-fn}{v} = \\sin \\theta \\implies \\htmlData{anchor=dv-deriv}{\\frac{dv}{d\\theta}} = \\cos \\theta$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"du-deriv"},{"color":"amber","style":"text-color","anchor":"dv-deriv"}],"connections":[{"to":"du-deriv","from":"u-fn","color":"sky","label":"differentiate","style":"solid"},{"to":"dv-deriv","from":"v-fn","color":"amber","label":"differentiate","style":"solid"}]},"order":1,"title":"Differentiate component functions","description":"Next, we differentiate both component functions with respect to $\\theta$:\n\n- For $u$, we apply the power rule: $\\frac{du}{d\\theta} = 12\\theta^2 + 5$\n- For $v$, we recall that the derivative of $\\sin \\theta$ is $\\cos \\theta$: $\\frac{dv}{d\\theta} = \\cos \\theta$\n\n*(This earns the accuracy mark **A1**)*"},{"type":"text","order":2,"title":"Recall the quotient rule","content":"The **Quotient Rule** formula is given by:\n\n$$\\frac{d\\tau}{d\\theta} = \\frac{v \\frac{du}{d\\theta} - u \\frac{dv}{d\\theta}}{v^2}$$\n\nThis formula can be found in your formula booklet. Make sure to subtract the terms in the correct order in the numerator.\n\n*(This step earns the method mark **M1**)*"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\frac{d\\tau}{d\\theta} = \\frac{\\htmlData{anchor=v-sub}{\\color{@amber}{\\sin \\theta}}\\htmlData{anchor=du-sub}{\\color{@sky}{(12\\theta^2 + 5)}} - \\htmlData{anchor=u-sub}{(4\\theta^3 + 5\\theta)}\\htmlData{anchor=dv-sub}{\\color{@purple}{\\cos \\theta}}}{\\htmlData{anchor=v2-sub}{\\color{@amber}{(\\sin \\theta)^2}}}$"},{"latex":"$$\\phantom{\\frac{d\\tau}{d\\theta}} = \\frac{(12\\theta^2 + 5)\\sin \\theta - (4\\theta^3 + 5\\theta)\\cos \\theta}{\\sin^2 \\theta}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"du-sub"},{"color":"purple","style":"text-color","anchor":"dv-sub"}],"connections":[{"to":"v2-sub","from":"v-sub","color":"amber","label":null,"style":"dashed"}]},"order":3,"title":"Substitute and simplify","description":"Substitute our expressions for $u$, $v$, $\\frac{du}{d\\theta}$, and $\\frac{dv}{d\\theta}$ into the formula:\n\n- $v \\frac{du}{d\\theta} = (\\sin \\theta)(12\\theta^2 + 5)$\n- $u \\frac{dv}{d\\theta} = (4\\theta^3 + 5\\theta)(\\cos \\theta)$\n- $v^2 = (\\sin \\theta)^2 = \\sin^2 \\theta$\n\nPutting it all together, we get our final expression.\n\n*(This final correct substitution and simplification earns the accuracy mark **A1**)*"}]},{"id":"product_rule","label":"Product Rule","steps":[{"type":"text","order":0,"title":"Rewrite as a product","content":"To apply the product rule, we first rewrite the quotient as a product by raising the denominator to the power of $-1$:\n\n$$\\tau(\\theta) = (4\\theta^3 + 5\\theta)(\\sin \\theta)^{-1}$$\n\nThis is now in the form $\\tau(\\theta) = u(\\theta)v(\\theta)$, where we define the two component functions as:\n* $$u(\\theta) = 4\\theta^3 + 5\\theta$$\n* $$v(\\theta) = (\\sin \\theta)^{-1}$$\n\nThis sets us up to use the product rule: \n$$\\frac{d\\tau}{d\\theta} = u'v + uv'$$"},{"type":"text","order":1,"title":"Differentiate component functions","content":"Now we find the derivative of each component function with respect to $\\theta$.\n\nFor $u(\\theta) = 4\\theta^3 + 5\\theta$, we use the power rule:\n$$u'(\\theta) = 12\\theta^2 + 5$$\n\nFor $v(\\theta) = (\\sin \\theta)^{-1}$, we apply the **chain rule**:\n$$\\begin{aligned} v'(\\theta) &= -1(\\sin \\theta)^{-2} \\cdot \\frac{d}{d\\theta}(\\sin \\theta) \\\\ &= -(\\sin \\theta)^{-2} \\cos \\theta \\\\ &= -\\frac{\\cos \\theta}{\\sin^2 \\theta} \\end{aligned}$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$$u = \\htmlData{anchor=u-func}{\\color{@sky}{4\\theta^3 + 5\\theta}}, \\quad u' = \\htmlData{anchor=u-deriv}{\\color{@pink}{12\\theta^2 + 5}}$$"},{"latex":"$$$v = \\htmlData{anchor=v-func}{\\color{@purple}{(\\sin \\theta)^{-1}}}, \\quad v' = \\htmlData{anchor=v-deriv}{\\color{@amber}{-(\\sin \\theta)^{-2} \\cos \\theta}}$$"},{"latex":"$$$\\frac{d\\tau}{d\\theta} = \\htmlData{anchor=term1}{\\color{@pink}{(12\\theta^2 + 5)}}\\htmlData{anchor=term2}{\\color{@purple}{(\\sin \\theta)^{-1}}} + \\htmlData{anchor=term3}{\\color{@sky}{(4\\theta^3 + 5\\theta)}}\\htmlData{anchor=term4}{\\color{@amber}{\\left(-(\\sin \\theta)^{-2} \\cos \\theta\\right)}}$$"}],"highlights":[{"color":"pink","style":"text-color","anchor":"term1"},{"color":"purple","style":"text-color","anchor":"term2"},{"color":"sky","style":"text-color","anchor":"term3"},{"color":"amber","style":"text-color","anchor":"term4"}],"connections":[{"to":"term1","from":"u-deriv","color":"pink","label":"u'","style":"solid"},{"to":"term2","from":"v-func","color":"purple","label":"v","style":"solid"},{"to":"term3","from":"u-func","color":"sky","label":"u","style":"solid"},{"to":"term4","from":"v-deriv","color":"amber","label":"v'","style":"solid"}]},"order":2,"title":"Substitute into product rule","description":"We substitute our functions and their derivatives into the product rule formula: $\\frac{d\\tau}{d\\theta} = u'v + uv'$."},{"type":"text","order":3,"title":"Simplify the final expression","content":"To write our answer over a common denominator, we simplify the terms:\n\n$$\\frac{d\\tau}{d\\theta} = \\frac{12\\theta^2 + 5}{\\sin \\theta} - \\frac{(4\\theta^3 + 5\\theta)\\cos \\theta}{\\sin^2 \\theta}$$\n\nMultiplying the numerator and denominator of the first term by $\\sin \\theta$ to find a common denominator of $\\sin^2 \\theta$, we get:\n\n$$\\frac{d\\tau}{d\\theta} = \\frac{(12\\theta^2 + 5)\\sin \\theta - (4\\theta^3 + 5\\theta)\\cos \\theta}{\\sin^2 \\theta}$$\n\nThis matches the standard quotient rule form required by the markscheme and earns full marks."}]}],"generatedAt":"2026-06-09T10:57:21.518Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"521a3e89-bc6f-472d-af92-1bd34422b69c:quotient_rule","label":"Quotient Rule","steps":[{"type":"text","order":0,"title":"Identify quotient components","content":"To find the rate of change of torque, $\\frac{d\\tau}{d\\theta}$, we can apply the **Quotient Rule** because the function $\\tau(\\theta)$ is expressed as one function divided by another:\n\n$$\\tau(\\theta) = \\frac{u}{v} = \\frac{4\\theta^3 + 5\\theta}{\\sin \\theta}$$\n\nFirst, let's identify our numerator, $u$, and our denominator, $v$:\n\n$$u = 4\\theta^3 + 5\\theta$$\n$$v = \\sin \\theta$$\n\n*(This step earns the first method mark **M1**)*","highlights":[{"text":"\\tau(\\theta) = \\frac{4\\theta^3 + 5\\theta}{\\sin \\theta}","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\text{Numerator: } \\htmlData{anchor=u-fn}{u} = 4\\theta^3 + 5\\theta \\implies \\htmlData{anchor=du-deriv}{\\frac{du}{d\\theta}} = 12\\theta^2 + 5$"},{"latex":"$$\\text{Denominator: } \\htmlData{anchor=v-fn}{v} = \\sin \\theta \\implies \\htmlData{anchor=dv-deriv}{\\frac{dv}{d\\theta}} = \\cos \\theta$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"du-deriv"},{"color":"amber","style":"text-color","anchor":"dv-deriv"}],"connections":[{"to":"du-deriv","from":"u-fn","color":"sky","label":"differentiate","style":"solid"},{"to":"dv-deriv","from":"v-fn","color":"amber","label":"differentiate","style":"solid"}]},"order":1,"title":"Differentiate component functions","description":"Next, we differentiate both component functions with respect to $\\theta$:\n\n- For $u$, we apply the power rule: $\\frac{du}{d\\theta} = 12\\theta^2 + 5$\n- For $v$, we recall that the derivative of $\\sin \\theta$ is $\\cos \\theta$: $\\frac{dv}{d\\theta} = \\cos \\theta$\n\n*(This earns the accuracy mark **A1**)*"},{"type":"text","order":2,"title":"Recall the quotient rule","content":"The **Quotient Rule** formula is given by:\n\n$$\\frac{d\\tau}{d\\theta} = \\frac{v \\frac{du}{d\\theta} - u \\frac{dv}{d\\theta}}{v^2}$$\n\nThis formula can be found in your formula booklet. Make sure to subtract the terms in the correct order in the numerator.\n\n*(This step earns the method mark **M1**)*"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\frac{d\\tau}{d\\theta} = \\frac{\\htmlData{anchor=v-sub}{\\color{@amber}{\\sin \\theta}}\\htmlData{anchor=du-sub}{\\color{@sky}{(12\\theta^2 + 5)}} - \\htmlData{anchor=u-sub}{(4\\theta^3 + 5\\theta)}\\htmlData{anchor=dv-sub}{\\color{@purple}{\\cos \\theta}}}{\\htmlData{anchor=v2-sub}{\\color{@amber}{(\\sin \\theta)^2}}}$"},{"latex":"$$\\phantom{\\frac{d\\tau}{d\\theta}} = \\frac{(12\\theta^2 + 5)\\sin \\theta - (4\\theta^3 + 5\\theta)\\cos \\theta}{\\sin^2 \\theta}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"du-sub"},{"color":"purple","style":"text-color","anchor":"dv-sub"}],"connections":[{"to":"v2-sub","from":"v-sub","color":"amber","label":null,"style":"dashed"}]},"order":3,"title":"Substitute and simplify","description":"Substitute our expressions for $u$, $v$, $\\frac{du}{d\\theta}$, and $\\frac{dv}{d\\theta}$ into the formula:\n\n- $v \\frac{du}{d\\theta} = (\\sin \\theta)(12\\theta^2 + 5)$\n- $u \\frac{dv}{d\\theta} = (4\\theta^3 + 5\\theta)(\\cos \\theta)$\n- $v^2 = (\\sin \\theta)^2 = \\sin^2 \\theta$\n\nPutting it all together, we get our final expression.\n\n*(This final correct substitution and simplification earns the accuracy mark **A1**)*"}]},{"id":"521a3e89-bc6f-472d-af92-1bd34422b69c:product_rule","label":"Product Rule","steps":[{"type":"text","order":0,"title":"Rewrite as a product","content":"To apply the product rule, we first rewrite the quotient as a product by raising the denominator to the power of $-1$:\n\n$$\\tau(\\theta) = (4\\theta^3 + 5\\theta)(\\sin \\theta)^{-1}$$\n\nThis is now in the form $\\tau(\\theta) = u(\\theta)v(\\theta)$, where we define the two component functions as:\n* $$u(\\theta) = 4\\theta^3 + 5\\theta$$\n* $$v(\\theta) = (\\sin \\theta)^{-1}$$\n\nThis sets us up to use the product rule: \n$$\\frac{d\\tau}{d\\theta} = u'v + uv'$$"},{"type":"text","order":1,"title":"Differentiate component functions","content":"Now we find the derivative of each component function with respect to $\\theta$.\n\nFor $u(\\theta) = 4\\theta^3 + 5\\theta$, we use the power rule:\n$$u'(\\theta) = 12\\theta^2 + 5$$\n\nFor $v(\\theta) = (\\sin \\theta)^{-1}$, we apply the **chain rule**:\n$$\\begin{aligned} v'(\\theta) &= -1(\\sin \\theta)^{-2} \\cdot \\frac{d}{d\\theta}(\\sin \\theta) \\\\ &= -(\\sin \\theta)^{-2} \\cos \\theta \\\\ &= -\\frac{\\cos \\theta}{\\sin^2 \\theta} \\end{aligned}$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$$u = \\htmlData{anchor=u-func}{\\color{@sky}{4\\theta^3 + 5\\theta}}, \\quad u' = \\htmlData{anchor=u-deriv}{\\color{@pink}{12\\theta^2 + 5}}$$"},{"latex":"$$$v = \\htmlData{anchor=v-func}{\\color{@purple}{(\\sin \\theta)^{-1}}}, \\quad v' = \\htmlData{anchor=v-deriv}{\\color{@amber}{-(\\sin \\theta)^{-2} \\cos \\theta}}$$"},{"latex":"$$$\\frac{d\\tau}{d\\theta} = \\htmlData{anchor=term1}{\\color{@pink}{(12\\theta^2 + 5)}}\\htmlData{anchor=term2}{\\color{@purple}{(\\sin \\theta)^{-1}}} + \\htmlData{anchor=term3}{\\color{@sky}{(4\\theta^3 + 5\\theta)}}\\htmlData{anchor=term4}{\\color{@amber}{\\left(-(\\sin \\theta)^{-2} \\cos \\theta\\right)}}$$"}],"highlights":[{"color":"pink","style":"text-color","anchor":"term1"},{"color":"purple","style":"text-color","anchor":"term2"},{"color":"sky","style":"text-color","anchor":"term3"},{"color":"amber","style":"text-color","anchor":"term4"}],"connections":[{"to":"term1","from":"u-deriv","color":"pink","label":"u'","style":"solid"},{"to":"term2","from":"v-func","color":"purple","label":"v","style":"solid"},{"to":"term3","from":"u-func","color":"sky","label":"u","style":"solid"},{"to":"term4","from":"v-deriv","color":"amber","label":"v'","style":"solid"}]},"order":2,"title":"Substitute into product rule","description":"We substitute our functions and their derivatives into the product rule formula: $\\frac{d\\tau}{d\\theta} = u'v + uv'$."},{"type":"text","order":3,"title":"Simplify the final expression","content":"To write our answer over a common denominator, we simplify the terms:\n\n$$\\frac{d\\tau}{d\\theta} = \\frac{12\\theta^2 + 5}{\\sin \\theta} - \\frac{(4\\theta^3 + 5\\theta)\\cos \\theta}{\\sin^2 \\theta}$$\n\nMultiplying the numerator and denominator of the first term by $\\sin \\theta$ to find a common denominator of $\\sin^2 \\theta$, we get:\n\n$$\\frac{d\\tau}{d\\theta} = \\frac{(12\\theta^2 + 5)\\sin \\theta - (4\\theta^3 + 5\\theta)\\cos \\theta}{\\sin^2 \\theta}$$\n\nThis matches the standard quotient rule form required by the markscheme and earns full marks."}]}],"generatedAt":"2026-06-09T10:57:21.587Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":null,"walkthroughVideo":null},{"id":"1dd6e7ed-234f-4728-8cdb-9ab46caf8736","revisionId":"c841e1cc-5490-49c4-86fd-bbea13a06cd3","questionNumber":null,"questionType":"LA","specification":"","difficultyBand":"easy","tier":null,"totalMarks":3,"sourceTitle":null,"calculatorUse":null,"isCalculation":false,"parts":[{"id":"4f67d304-57e3-4673-b38c-baa454497e28","label":null,"content":"Given that \\[y = \\frac{x^4}{\\cos x}\\] find \$\\frac{\\text{d}y}{\\text{d}x}\$.","metadata":null,"markscheme":"- Identifies $u = x^4 \\Rightarrow \\frac{du}{dx} = 4x^3$ and $v = \\cos x \\Rightarrow \\frac{dv}{dx} = -\\sin x$ M1\n- Substitutes these derivatives into the quotient rule formula to obtain $\\frac{dy}{dx} = \\frac{(\\cos x)(4x^3) - (x^4)(-\\sin x)}{(\\cos x)^2}$ M1\n- Simplifies the numerator to reach the final expression $\\frac{4x^3 \\cos x + x^4 \\sin x}{\\cos^2 x}$ A1","marks":3,"order":0,"markingRubric":null,"tableImages":[],"explanation":{"methods":[{"id":"quotient_rule","label":"Quotient Rule","steps":[{"type":"text","order":0,"title":"Identify the quotient rule","content":"We need to find the derivative of the function:\n\n$$y = \\frac{x^4}{\\cos x}$$\n\nSince this function is in the form of a fraction where both the numerator and denominator are functions of $x$, we use the **Quotient Rule**. From the data booklet, the quotient rule is:\n\n$$\\text{If } y = \\frac{u}{v}, \\text{ then } \\frac{\\text{d}y}{\\text{d}x} = \\frac{v\\frac{\\text{d}u}{\\text{d}x} - u\\frac{\\text{d}v}{\\text{d}x}}{v^2}$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$y = \\frac{\\htmlData{anchor=u-expr}{\\color{@sky}{x^4}}}{\\htmlData{anchor=v-expr}{\\color{@amber}{\\cos x}}}$"},{"latex":"$$\\htmlData{anchor=u-def}{\\color{@sky}{u = x^4}} \\implies \\htmlData{anchor=du-def}{\\color{@purple}{\\frac{\\text{d}u}{\\text{d}x} = 4x^3}}$"},{"latex":"$$\\htmlData{anchor=v-def}{\\color{@amber}{v = \\cos x}} \\implies \\htmlData{anchor=dv-def}{\\color{@teal}{\\frac{\\text{d}v}{\\text{d}x} = -\\sin x}}$"}],"highlights":[{"color":"sky","style":"underline","anchor":"u-def"},{"color":"amber","style":"underline","anchor":"v-def"}],"connections":[{"to":"u-def","from":"u-expr","color":"sky","label":null,"style":"solid"},{"to":"v-def","from":"v-expr","color":"amber","label":null,"style":"solid"}]},"order":1,"title":"Define $u$ and $v$","description":"Identify the numerator $u$ and the denominator $v$, and differentiate each with respect to $x$ to find $\\frac{\\text{d}u}{\\text{d}x}$ and $\\frac{\\text{d}v}{\\text{d}x}$. This step earns you the first mark (**M1**)."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\frac{\\text{d}y}{\\text{d}x} = \\frac{\\htmlData{anchor=v-sub-eq}{\\color{@amber}{v}} \\cdot \\htmlData{anchor=du-sub-eq}{\\color{@purple}{\\frac{\\text{d}u}{\\text{d}x}}} - \\htmlData{anchor=u-sub-eq}{\\color{@sky}{u}} \\cdot \\htmlData{anchor=dv-sub-eq}{\\color{@teal}{\\frac{\\text{d}v}{\\text{d}x}}}}{\\htmlData{anchor=v2-sub-eq}{\\color{@amber}{v^2}}}$"},{"latex":"$$\\phantom{\\frac{\\text{d}y}{\\text{d}x}} = \\frac{(\\htmlData{anchor=v-val}{\\color{@amber}{\\cos x}})(\\htmlData{anchor=du-val}{\\color{@purple}{4x^3}}) - (\\htmlData{anchor=u-val}{\\color{@sky}{x^4}})(\\htmlData{anchor=dv-val}{\\color{@teal}{-\\sin x}})}{(\\htmlData{anchor=v2-val}{\\color{@amber}{\\cos x}})^2}$"}],"highlights":[{"color":"amber","style":"underline","anchor":"v-val"},{"color":"purple","style":"underline","anchor":"du-val"}],"connections":[{"to":"v-val","from":"v-sub-eq","color":"amber","label":null,"style":"solid"},{"to":"du-val","from":"du-sub-eq","color":"purple","label":null,"style":"solid"},{"to":"u-val","from":"u-sub-eq","color":"sky","label":null,"style":"solid"},{"to":"dv-val","from":"dv-sub-eq","color":"teal","label":null,"style":"solid"}]},"order":2,"title":"Apply quotient rule","description":"Substitute $u$, $v$, $\\frac{\\text{d}u}{\\text{d}x}$, and $\\frac{\\text{d}v}{\\text{d}x}$ into the quotient rule formula. This substitution earns you the second mark (**M1**)."},{"type":"text","order":3,"title":"Simplify final expression","content":"Now, we simplify the terms in the numerator:\n\n* The first term is $(\\cos x)(4x^3) = 4x^3 \\cos x$\n* The second term is $- (x^4)(-\\sin x) = + x^4 \\sin x$\n* The denominator is $(\\cos x)^2 = \\cos^2 x$\n\nPutting it all together, we get our final simplified answer:\n\n$$\\frac{\\text{d}y}{\\text{d}x} = \\frac{4x^3 \\cos x + x^4 \\sin x}{\\cos^2 x}$$\n\nThis final simplification earns you the accuracy mark (**A1**)."}]},{"id":"product_rule","label":"Product Rule","steps":[{"type":"text","order":0,"title":"Rewrite as a product","content":"To differentiate the given function using the product rule, we first rewrite the quotient as a product of two terms: $$y = x^4 (\\cos x)^{-1}$$ Now, we can define our two functions: $$u = x^4$$ and $$v = (\\cos x)^{-1}$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$u = x^4 \\implies \\htmlData{anchor=du-val}{\\frac{du}{dx} = 4x^3}$"},{"latex":"$$v = (\\cos x)^{-1} \\implies \\htmlData{anchor=dv-val}{\\frac{dv}{dx} = -1(\\cos x)^{-2} \\cdot (-\\sin x) = \\frac{\\sin x}{\\cos^2 x}}$"}],"highlights":[{"color":"sky","style":"box","anchor":"du-val"},{"color":"pink","style":"box","anchor":"dv-val"}],"connections":null},"order":1,"title":"Find the individual derivatives","description":"We differentiate $u$ normally using the power rule, and we differentiate $v$ using the chain rule."},{"type":"text","order":2,"title":"Apply the product rule","content":"The product rule states that: $$\\frac{dy}{dx} = u\\frac{dv}{dx} + v\\frac{du}{dx}$$ Substituting our functions and their derivatives into this formula, we get: $$\\frac{dy}{dx} = (x^4)\\left(\\frac{\\sin x}{\\cos^2 x}\\right) + (\\cos x)^{-1}(4x^3)$$ This simplifies to: $$\\frac{dy}{dx} = \\frac{x^4 \\sin x}{\\cos^2 x} + \\frac{4x^3}{\\cos x}$$"},{"type":"text","order":3,"title":"Simplify with common denominator","content":"To obtain the final simplified expression, we express both terms over a common denominator of $\\cos^2 x$. Multiplying the numerator and denominator of the second term by $\\cos x$ gives: $$\\frac{dy}{dx} = \\frac{x^4 \\sin x}{\\cos^2 x} + \\frac{4x^3 \\cos x}{\\cos^2 x}$$ Combining these over the single denominator yields the final answer: $$\\frac{dy}{dx} = \\frac{4x^3 \\cos x + x^4 \\sin x}{\\cos^2 x}$$"}]},{"id":"logarithmic_differentiation","label":"Logarithmic Differentiation","steps":[{"type":"text","order":0,"title":"Introduce Logarithmic Differentiation","content":"To differentiate $y = \\frac{x^4}{\\cos x}$ using logarithmic differentiation, we begin by taking the natural logarithm ($\\ln$) of both sides of the equation:\n\n$$\\ln(y) = \\ln\\left(\\frac{x^4}{\\cos x}\\right)$$\n\nThis method is particularly useful here as it allows us to convert the quotient into simpler, separate terms before differentiating."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\ln(y) = \\htmlData{anchor=log-frac}{\\ln\\left(\\frac{x^4}{\\cos x}\\right)}$"},{"latex":"$$\\ln(y) = \\htmlData{anchor=term1}{\\ln(x^4)} - \\htmlData{anchor=term2}{\\ln(\\cos x)}$"},{"latex":"$$\\ln(y) = \\htmlData{anchor=final-exp}{4\\ln(x)} - \\ln(\\cos x)$"}],"highlights":[{"color":"sky","style":"box","anchor":"log-frac"},{"color":"sky","style":"text-color","anchor":"term1"},{"color":"pink","style":"text-color","anchor":"term2"},{"color":"amber","style":"text-color","anchor":"final-exp"}],"connections":[{"to":"term1","from":"log-frac","color":"sky","label":"log laws","style":"solid"},{"to":"final-exp","from":"term1","color":"amber","label":"power rule","style":"solid"}]},"order":1,"title":"Expand using log laws","description":"We use the logarithm laws $\\ln\\left(\\frac{a}{b}\\right) = \\ln(a) - \\ln(b)$ and $\\ln(a^k) = k\\ln(a)$ to expand the right-hand side."},{"type":"text","order":2,"title":"Differentiate implicitly","content":"Now we differentiate both sides of the equation with respect to $x$.\n\nOn the left-hand side, we apply the chain rule (implicit differentiation):\n$$\\frac{d}{dx}[\\ln(y)] = \\frac{1}{y}\\frac{dy}{dx}$$\n\nOn the right-hand side, we differentiate term-by-term:\n- The derivative of $4\\ln(x)$ is $\\frac{4}{x}$.\n- For $\\ln(\\cos x)$, we use the chain rule:\n $$\\frac{d}{dx}[\\ln(\\cos x)] = \\frac{1}{\\cos x} \\cdot (-\\sin x) = -\\tan x$$\n\nPutting this together, we get:\n$$\\frac{1}{y}\\frac{dy}{dx} = \\frac{4}{x} - (-\\tan x) = \\frac{4}{x} + \\frac{\\sin x}{\\cos x}$$"},{"type":"text","order":3,"title":"Solve for dy/dx","content":"Multiply both sides of the equation by $y$ to isolate $\\frac{dy}{dx}$:\n\n$$\\frac{dy}{dx} = y \\left(\\frac{4}{x} + \\frac{\\sin x}{\\cos x}\\right)$$\n\nNow, substitute the original expression for $y = \\frac{x^4}{\\cos x}$ back into this equation:\n\n$$\\frac{dy}{dx} = \\left(\\frac{x^4}{\\cos x}\\right)\\left(\\frac{4}{x} + \\frac{\\sin x}{\\cos x}\\right)$$\n\nExpanding the brackets gives:\n\n$$\\frac{dy}{dx} = \\frac{4x^3}{\\cos x} + \\frac{x^4\\sin x}{\\cos^2 x}$$"},{"type":"text","order":4,"title":"Simplify the final fraction","content":"To match standard forms, we write the terms over a common denominator of $\\cos^2 x$:\n\n$$\\frac{dy}{dx} = \\frac{4x^3\\cos x}{\\cos^2 x} + \\frac{x^4\\sin x}{\\cos^2 x}$$\n\nCombining these into a single fraction gives the final derivative:\n\n$$\\frac{dy}{dx} = \\frac{4x^3\\cos x + x^4\\sin x}{\\cos^2 x}$$"}]}],"generatedAt":"2026-06-09T03:42:27.203Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"4f67d304-57e3-4673-b38c-baa454497e28:quotient_rule","label":"Quotient Rule","steps":[{"type":"text","order":0,"title":"Identify the quotient rule","content":"We need to find the derivative of the function:\n\n$$y = \\frac{x^4}{\\cos x}$$\n\nSince this function is in the form of a fraction where both the numerator and denominator are functions of $x$, we use the **Quotient Rule**. From the data booklet, the quotient rule is:\n\n$$\\text{If } y = \\frac{u}{v}, \\text{ then } \\frac{\\text{d}y}{\\text{d}x} = \\frac{v\\frac{\\text{d}u}{\\text{d}x} - u\\frac{\\text{d}v}{\\text{d}x}}{v^2}$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$y = \\frac{\\htmlData{anchor=u-expr}{\\color{@sky}{x^4}}}{\\htmlData{anchor=v-expr}{\\color{@amber}{\\cos x}}}$"},{"latex":"$$\\htmlData{anchor=u-def}{\\color{@sky}{u = x^4}} \\implies \\htmlData{anchor=du-def}{\\color{@purple}{\\frac{\\text{d}u}{\\text{d}x} = 4x^3}}$"},{"latex":"$$\\htmlData{anchor=v-def}{\\color{@amber}{v = \\cos x}} \\implies \\htmlData{anchor=dv-def}{\\color{@teal}{\\frac{\\text{d}v}{\\text{d}x} = -\\sin x}}$"}],"highlights":[{"color":"sky","style":"underline","anchor":"u-def"},{"color":"amber","style":"underline","anchor":"v-def"}],"connections":[{"to":"u-def","from":"u-expr","color":"sky","label":null,"style":"solid"},{"to":"v-def","from":"v-expr","color":"amber","label":null,"style":"solid"}]},"order":1,"title":"Define $u$ and $v$","description":"Identify the numerator $u$ and the denominator $v$, and differentiate each with respect to $x$ to find $\\frac{\\text{d}u}{\\text{d}x}$ and $\\frac{\\text{d}v}{\\text{d}x}$. This step earns you the first mark (**M1**)."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\frac{\\text{d}y}{\\text{d}x} = \\frac{\\htmlData{anchor=v-sub-eq}{\\color{@amber}{v}} \\cdot \\htmlData{anchor=du-sub-eq}{\\color{@purple}{\\frac{\\text{d}u}{\\text{d}x}}} - \\htmlData{anchor=u-sub-eq}{\\color{@sky}{u}} \\cdot \\htmlData{anchor=dv-sub-eq}{\\color{@teal}{\\frac{\\text{d}v}{\\text{d}x}}}}{\\htmlData{anchor=v2-sub-eq}{\\color{@amber}{v^2}}}$"},{"latex":"$$\\phantom{\\frac{\\text{d}y}{\\text{d}x}} = \\frac{(\\htmlData{anchor=v-val}{\\color{@amber}{\\cos x}})(\\htmlData{anchor=du-val}{\\color{@purple}{4x^3}}) - (\\htmlData{anchor=u-val}{\\color{@sky}{x^4}})(\\htmlData{anchor=dv-val}{\\color{@teal}{-\\sin x}})}{(\\htmlData{anchor=v2-val}{\\color{@amber}{\\cos x}})^2}$"}],"highlights":[{"color":"amber","style":"underline","anchor":"v-val"},{"color":"purple","style":"underline","anchor":"du-val"}],"connections":[{"to":"v-val","from":"v-sub-eq","color":"amber","label":null,"style":"solid"},{"to":"du-val","from":"du-sub-eq","color":"purple","label":null,"style":"solid"},{"to":"u-val","from":"u-sub-eq","color":"sky","label":null,"style":"solid"},{"to":"dv-val","from":"dv-sub-eq","color":"teal","label":null,"style":"solid"}]},"order":2,"title":"Apply quotient rule","description":"Substitute $u$, $v$, $\\frac{\\text{d}u}{\\text{d}x}$, and $\\frac{\\text{d}v}{\\text{d}x}$ into the quotient rule formula. This substitution earns you the second mark (**M1**)."},{"type":"text","order":3,"title":"Simplify final expression","content":"Now, we simplify the terms in the numerator:\n\n* The first term is $(\\cos x)(4x^3) = 4x^3 \\cos x$\n* The second term is $- (x^4)(-\\sin x) = + x^4 \\sin x$\n* The denominator is $(\\cos x)^2 = \\cos^2 x$\n\nPutting it all together, we get our final simplified answer:\n\n$$\\frac{\\text{d}y}{\\text{d}x} = \\frac{4x^3 \\cos x + x^4 \\sin x}{\\cos^2 x}$$\n\nThis final simplification earns you the accuracy mark (**A1**)."}]},{"id":"4f67d304-57e3-4673-b38c-baa454497e28:product_rule","label":"Product Rule","steps":[{"type":"text","order":0,"title":"Rewrite as a product","content":"To differentiate the given function using the product rule, we first rewrite the quotient as a product of two terms: $$y = x^4 (\\cos x)^{-1}$$ Now, we can define our two functions: $$u = x^4$$ and $$v = (\\cos x)^{-1}$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$u = x^4 \\implies \\htmlData{anchor=du-val}{\\frac{du}{dx} = 4x^3}$"},{"latex":"$$v = (\\cos x)^{-1} \\implies \\htmlData{anchor=dv-val}{\\frac{dv}{dx} = -1(\\cos x)^{-2} \\cdot (-\\sin x) = \\frac{\\sin x}{\\cos^2 x}}$"}],"highlights":[{"color":"sky","style":"box","anchor":"du-val"},{"color":"pink","style":"box","anchor":"dv-val"}],"connections":null},"order":1,"title":"Find the individual derivatives","description":"We differentiate $u$ normally using the power rule, and we differentiate $v$ using the chain rule."},{"type":"text","order":2,"title":"Apply the product rule","content":"The product rule states that: $$\\frac{dy}{dx} = u\\frac{dv}{dx} + v\\frac{du}{dx}$$ Substituting our functions and their derivatives into this formula, we get: $$\\frac{dy}{dx} = (x^4)\\left(\\frac{\\sin x}{\\cos^2 x}\\right) + (\\cos x)^{-1}(4x^3)$$ This simplifies to: $$\\frac{dy}{dx} = \\frac{x^4 \\sin x}{\\cos^2 x} + \\frac{4x^3}{\\cos x}$$"},{"type":"text","order":3,"title":"Simplify with common denominator","content":"To obtain the final simplified expression, we express both terms over a common denominator of $\\cos^2 x$. Multiplying the numerator and denominator of the second term by $\\cos x$ gives: $$\\frac{dy}{dx} = \\frac{x^4 \\sin x}{\\cos^2 x} + \\frac{4x^3 \\cos x}{\\cos^2 x}$$ Combining these over the single denominator yields the final answer: $$\\frac{dy}{dx} = \\frac{4x^3 \\cos x + x^4 \\sin x}{\\cos^2 x}$$"}]},{"id":"4f67d304-57e3-4673-b38c-baa454497e28:logarithmic_differentiation","label":"Logarithmic Differentiation","steps":[{"type":"text","order":0,"title":"Introduce Logarithmic Differentiation","content":"To differentiate $y = \\frac{x^4}{\\cos x}$ using logarithmic differentiation, we begin by taking the natural logarithm of the absolute value of both sides. For $x \\ne 0$ and $\\cos x \\ne 0$, we have $y \\ne 0$, so:\n\n$$\\ln|y| = \\ln\\left|\\frac{x^4}{\\cos x}\\right|$$\n\nThe absolute values ensure that the logarithms are defined when $x$ or $y$ is negative. The case $x=0$, which is excluded by this method, will be checked separately."},{"type":"hyper_latex","block":{"nodes":[{"latex":"\\ln|y| = \\htmlData{anchor=log-frac}{\\ln\\left|\\frac{x^4}{\\cos x}\\right|}"},{"latex":"\\ln|y| = \\htmlData{anchor=term1}{\\ln(x^4)} - \\htmlData{anchor=term2}{\\ln|\\cos x|}"},{"latex":"\\ln|y| = \\htmlData{anchor=final-exp}{4\\ln|x|} - \\ln|\\cos x|"}],"highlights":[{"color":"sky","style":"box","anchor":"log-frac"},{"color":"sky","style":"text-color","anchor":"term1"},{"color":"pink","style":"text-color","anchor":"term2"},{"color":"amber","style":"text-color","anchor":"final-exp"}],"connections":[{"to":"term1","from":"log-frac","color":"sky","label":"log laws","style":"solid"},{"to":"final-exp","from":"term1","color":"amber","label":"power rule","style":"solid"}]},"order":1,"title":"Expand using log laws","description":"We use the logarithm laws $\\ln\\left|\\frac{a}{b}\\right| = \\ln|a| - \\ln|b|$ and, for $x \\ne 0$, $\\ln(x^4) = 4\\ln|x|$ to expand the right-hand side."},{"type":"text","order":2,"title":"Differentiate implicitly","content":"Now we differentiate both sides of the equation with respect to $x$, for $x \\ne 0$ and $\\cos x \\ne 0$.\n\nOn the left-hand side, we apply the chain rule (implicit differentiation):\n$$\\frac{d}{dx}[\\ln|y|] = \\frac{1}{y}\\frac{dy}{dx}$$\n\nOn the right-hand side, we differentiate term-by-term:\n- The derivative of $4\\ln|x|$ is $\\frac{4}{x}$.\n- For $\\ln|\\cos x|$, we use the chain rule:\n $$\\frac{d}{dx}[\\ln|\\cos x|] = \\frac{1}{\\cos x} \\cdot (-\\sin x) = -\\tan x$$\n\nPutting this together, we get:\n$$\\frac{1}{y}\\frac{dy}{dx} = \\frac{4}{x} - (-\\tan x) = \\frac{4}{x} + \\frac{\\sin x}{\\cos x}$$"},{"type":"text","order":3,"title":"Solve for dy/dx","content":"For $x \\ne 0$ and $\\cos x \\ne 0$, multiply both sides of the equation by $y$ to isolate $\\frac{dy}{dx}$:\n\n$$\\frac{dy}{dx} = y \\left(\\frac{4}{x} + \\frac{\\sin x}{\\cos x}\\right)$$\n\nNow, substitute the original expression for $y = \\frac{x^4}{\\cos x}$ back into this equation:\n\n$$\\frac{dy}{dx} = \\left(\\frac{x^4}{\\cos x}\\right)\\left(\\frac{4}{x} + \\frac{\\sin x}{\\cos x}\\right)$$\n\nExpanding the brackets gives:\n\n$$\\frac{dy}{dx} = \\frac{4x^3}{\\cos x} + \\frac{x^4\\sin x}{\\cos^2 x}$$"},{"type":"text","order":4,"title":"Simplify the final fraction","content":"For $x \\ne 0$ and $\\cos x \\ne 0$, we write the terms over a common denominator of $\\cos^2 x$:\n\n$$\\frac{dy}{dx} = \\frac{4x^3\\cos x}{\\cos^2 x} + \\frac{x^4\\sin x}{\\cos^2 x}$$\n\nCombining these into a single fraction gives:\n\n$$\\frac{dy}{dx} = \\frac{4x^3\\cos x + x^4\\sin x}{\\cos^2 x}$$\n\nIt remains to check $x=0$, since logarithmic differentiation excluded this value. From $y=x^4\\sec x$,\n$$y'(0)=\\lim_{h\\to 0}\\frac{h^4\\sec h}{h}=\\lim_{h\\to 0}h^3\\sec h=0.$$\nThe final expression also gives $0$ when $x=0$, so the result is valid for every $x$ such that $\\cos x\\ne 0$."}]}],"generatedAt":"2026-06-09T03:42:39.545Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":null,"walkthroughVideo":null},{"id":"ccae4139-d718-4443-8d1b-9e52e56a46fe","revisionId":"658f7505-72cb-45c7-925b-201162ea952f","questionNumber":null,"questionType":"LA","specification":"A robotic arm's radial extension \$s\$ is modeled by the function:\n\n\\[s = \\frac{3\\theta^6}{\\sin(4\\theta)}\\]\n\nwhere \$\\theta\$ is the joint angle in radians, for \$0 < \\theta < \\frac{\\pi}{4}\$.","difficultyBand":"easy","tier":null,"totalMarks":4,"sourceTitle":null,"calculatorUse":null,"isCalculation":true,"parts":[{"id":"200ac662-5bb1-4a8b-af41-adf83f72abac","label":null,"content":"Find the exact expression for \$\\frac{ds}{d\\theta}\$.","metadata":null,"markscheme":"- Identifies $u = 3\\theta^6$ and $v = \\sin(4\\theta)$ M1\n- Differentiates to find $\\frac{du}{d\\theta} = 18\\theta^5$ and $\\frac{dv}{d\\theta} = 4\\cos(4\\theta)$ (using the chain rule for the denominator) A1\n- Correctly applies the quotient rule formula $\\frac{v u' - u v'}{v^2}$ with their components: $\\frac{ds}{d\\theta} = \\frac{\\sin(4\\theta) \\cdot 18\\theta^5 - 3\\theta^6 \\cdot 4\\cos(4\\theta)}{(\\sin(4\\theta))^2}$ M1\n- Simplifies the expression to $\\frac{18\\theta^5 \\sin(4\\theta) - 12\\theta^6 \\cos(4\\theta)}{\\sin^2(4\\theta)}$ or an equivalent factorised form A1","marks":4,"order":0,"markingRubric":null,"tableImages":[],"explanation":{"methods":[{"id":"quotient_rule","label":"Quotient Rule","steps":[{"type":"text","order":0,"title":"Identify numerator and denominator","content":"To differentiate the function\n\n$$s = \\frac{3\\theta^6}{\\sin(4\\theta)}$$\n\nusing the **Quotient Rule**, we first define the numerator as $u$ and the denominator as $v$:\n\n$$u = 3\\theta^6$$\n$$v = \\sin(4\\theta)$$\n\nThis earns you the first mark (**M1**)."},{"type":"text","order":1,"title":"Differentiate both parts","content":"Next, we find the derivatives of $u$ and $v$ with respect to $\\theta$.\n\n- For $u = 3\\theta^6$, apply the power rule:\n $$\\frac{du}{d\\theta} = 18\\theta^5$$\n\n- For $v = \\sin(4\\theta)$, apply the chain rule:\n $$\\frac{dv}{d\\theta} = 4\\cos(4\\theta)$$\n\nThis step earns the second mark (**A1**)."},{"type":"hyper_latex","block":{"nodes":[{"text":"The Quotient Rule formula is:"},{"latex":"$$$\\frac{ds}{d\\theta} = \\frac{\\htmlData{anchor=v_term}{v} \\htmlData{anchor=du_term}{\\frac{du}{d\\theta}} - \\htmlData{anchor=u_term}{u} \\htmlData{anchor=dv_term}{\\frac{dv}{d\\theta}}}{\\htmlData{anchor=v2_term}{v^2}}$$"},{"text":"Substituting our terms into the formula:"},{"latex":"$$$\\frac{ds}{d\\theta} = \\frac{\\htmlData{anchor=v_sub}{\\color{@sky}{\\sin(4\\theta)}} \\cdot \\htmlData{anchor=du_sub}{\\color{@amber}{18\\theta^5}} - \\htmlData{anchor=u_sub}{\\color{@purple}{3\\theta^6}} \\cdot \\htmlData{anchor=dv_sub}{\\color{@teal}{4\\cos(4\\theta)}}}{\\htmlData{anchor=v2_sub}{\\color{@pink}{\\sin^2(4\\theta)}}}$$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"v_sub"},{"color":"amber","style":"text-color","anchor":"du_sub"},{"color":"purple","style":"text-color","anchor":"u_sub"},{"color":"teal","style":"text-color","anchor":"dv_sub"}],"connections":[{"to":"v_sub","from":"v_term","color":"sky","label":"v","style":"solid"},{"to":"du_sub","from":"du_term","color":"amber","label":"u'","style":"solid"},{"to":"u_sub","from":"u_term","color":"purple","label":"u","style":"solid"},{"to":"dv_sub","from":"dv_term","color":"teal","label":"v'","style":"solid"}]},"order":2,"title":"Substitute into Quotient Rule","description":"Now, we substitute our functions and their derivatives directly into the Quotient Rule formula. This step earns the third mark (**M1**)."},{"type":"text","order":3,"title":"Simplify the final expression","content":"Finally, simplify the numerator by multiplying the constant coefficients together:\n\n$$\\frac{ds}{d\\theta} = \\frac{18\\theta^5 \\sin(4\\theta) - 12\\theta^6 \\cos(4\\theta)}{\\sin^2(4\\theta)}$$\n\nAlternatively, you can factorise out common terms (such as $6\\theta^5$) from the numerator:\n\n$$\\frac{ds}{d\\theta} = \\frac{6\\theta^5 [3\\sin(4\\theta) - 2\\theta\\cos(4\\theta)]}{\\sin^2(4\\theta)}$$\n\nEither form is fully acceptable and earns the final accuracy mark (**A1**)."}]},{"id":"product_rule","label":"Product and Chain Rule","steps":[{"type":"text","order":0,"title":"Rewrite the function","content":"To use the product rule, we first rewrite the quotient $s = \\frac{3\\theta^6}{\\sin(4\\theta)}$ as a product by raising the denominator to the power of $-1$:\n\n$$s = 3\\theta^6 \\left(\\sin(4\\theta)\\right)^{-1}$$"},{"type":"text","order":1,"title":"Define product terms","content":"Now we identify the two functions $u$ and $v$ that make up our product $s = u \\cdot v$:\n\n$$u = 3\\theta^6 \\quad \\text{and} \\quad v = \\left(\\sin(4\\theta)\\right)^{-1}$$"},{"type":"text","order":2,"title":"Differentiate each part","content":"Next, we find the derivatives of $u$ and $v$ with respect to $\\theta$:\n\n1. For $u = 3\\theta^6$, using the power rule:\n$$\\frac{du}{d\\theta} = 18\\theta^5$$\n\n2. For $v = \\left(\\sin(4\\theta)\\right)^{-1}$, we use the chain rule:\n$$\\frac{dv}{d\\theta} = -1\\left(\\sin(4\\theta)\\right)^{-2} \\cdot 4\\cos(4\\theta) = -4\\cos(4\\theta)\\left(\\sin(4\\theta)\\right)^{-2}$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$u = \\htmlData{anchor=u-term}{3\\theta^6}, \\quad \\frac{du}{d\\theta} = \\htmlData{anchor=du-term}{18\\theta^5}$"},{"latex":"$$v = \\htmlData{anchor=v-term}{\\left(\\sin(4\\theta)\\right)^{-1}}, \\quad \\frac{dv}{d\\theta} = \\htmlData{anchor=dv-term}{-4\\cos(4\\theta)\\left(\\sin(4\\theta)\\right)^{-2}}$"},{"latex":"$$\\frac{ds}{d\\theta} = v \\frac{du}{d\\theta} + u \\frac{dv}{d\\theta}$"},{"latex":"$$\\frac{ds}{d\\theta} = \\htmlData{anchor=v-sub}{\\left(\\sin(4\\theta)\\right)^{-1}} \\cdot \\htmlData{anchor=du-sub}{18\\theta^5} + \\htmlData{anchor=u-sub}{3\\theta^6} \\cdot \\htmlData{anchor=dv-sub}{\\left(-4\\cos(4\\theta)\\left(\\sin(4\\theta)\\right)^{-2}\\right)}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"du-term"},{"color":"pink","style":"text-color","anchor":"v-term"},{"color":"purple","style":"text-color","anchor":"u-term"},{"color":"amber","style":"text-color","anchor":"dv-term"}],"connections":[{"to":"du-sub","from":"du-term","color":"sky","label":"du/dtheta","style":"solid"},{"to":"v-sub","from":"v-term","color":"pink","label":"v","style":"solid"},{"to":"u-sub","from":"u-term","color":"purple","label":"u","style":"solid"},{"to":"dv-sub","from":"dv-term","color":"amber","label":"dv/dtheta","style":"solid"}]},"order":3,"title":"Apply product rule","description":"Now we substitute these derivatives into the product rule formula: $$\\frac{ds}{d\\theta} = v \\frac{du}{d\\theta} + u \\frac{dv}{d\\theta}$$"},{"type":"text","order":4,"title":"Simplify final expression","content":"Finally, we simplify the terms and express them with a common denominator:\n\n$$\\frac{ds}{d\\theta} = \\frac{18\\theta^5}{\\sin(4\\theta)} - \\frac{12\\theta^6\\cos(4\\theta)}{\\sin^2(4\\theta)}$$\n\nBy multiplying the first term's numerator and denominator by $\\sin(4\\theta)$ to combine them over a common denominator, we get:\n\n$$\\frac{ds}{d\\theta} = \\frac{18\\theta^5\\sin(4\\theta) - 12\\theta^6\\cos(4\\theta)}{\\sin^2(4\\theta)}$$\n\nThis can also be factored as:\n\n$$\\frac{ds}{d\\theta} = \\frac{6\\theta^5\\left(3\\sin(4\\theta) - 2\\theta\\cos(4\\theta)\\right)}{\\sin^2(4\\theta)}$$"}]}],"generatedAt":"2026-06-09T09:11:16.376Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"200ac662-5bb1-4a8b-af41-adf83f72abac:quotient_rule","label":"Quotient Rule","steps":[{"type":"text","order":0,"title":"Identify numerator and denominator","content":"To differentiate the function\n\n$$s = \\frac{3\\theta^6}{\\sin(4\\theta)}$$\n\nusing the **Quotient Rule**, we first define the numerator as $u$ and the denominator as $v$:\n\n$$u = 3\\theta^6$$\n$$v = \\sin(4\\theta)$$\n\nThis earns you the first mark (**M1**)."},{"type":"text","order":1,"title":"Differentiate both parts","content":"Next, we find the derivatives of $u$ and $v$ with respect to $\\theta$.\n\n- For $u = 3\\theta^6$, apply the power rule:\n $$\\frac{du}{d\\theta} = 18\\theta^5$$\n\n- For $v = \\sin(4\\theta)$, apply the chain rule:\n $$\\frac{dv}{d\\theta} = 4\\cos(4\\theta)$$\n\nThis step earns the second mark (**A1**)."},{"type":"hyper_latex","block":{"nodes":[{"text":"The Quotient Rule formula is:"},{"latex":"$$$\\frac{ds}{d\\theta} = \\frac{\\htmlData{anchor=v_term}{v} \\htmlData{anchor=du_term}{\\frac{du}{d\\theta}} - \\htmlData{anchor=u_term}{u} \\htmlData{anchor=dv_term}{\\frac{dv}{d\\theta}}}{\\htmlData{anchor=v2_term}{v^2}}$$"},{"text":"Substituting our terms into the formula:"},{"latex":"$$$\\frac{ds}{d\\theta} = \\frac{\\htmlData{anchor=v_sub}{\\color{@sky}{\\sin(4\\theta)}} \\cdot \\htmlData{anchor=du_sub}{\\color{@amber}{18\\theta^5}} - \\htmlData{anchor=u_sub}{\\color{@purple}{3\\theta^6}} \\cdot \\htmlData{anchor=dv_sub}{\\color{@teal}{4\\cos(4\\theta)}}}{\\htmlData{anchor=v2_sub}{\\color{@pink}{\\sin^2(4\\theta)}}}$$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"v_sub"},{"color":"amber","style":"text-color","anchor":"du_sub"},{"color":"purple","style":"text-color","anchor":"u_sub"},{"color":"teal","style":"text-color","anchor":"dv_sub"}],"connections":[{"to":"v_sub","from":"v_term","color":"sky","label":"v","style":"solid"},{"to":"du_sub","from":"du_term","color":"amber","label":"u'","style":"solid"},{"to":"u_sub","from":"u_term","color":"purple","label":"u","style":"solid"},{"to":"dv_sub","from":"dv_term","color":"teal","label":"v'","style":"solid"}]},"order":2,"title":"Substitute into Quotient Rule","description":"Now, we substitute our functions and their derivatives directly into the Quotient Rule formula. This step earns the third mark (**M1**)."},{"type":"text","order":3,"title":"Simplify the final expression","content":"Finally, simplify the numerator by multiplying the constant coefficients together:\n\n$$\\frac{ds}{d\\theta} = \\frac{18\\theta^5 \\sin(4\\theta) - 12\\theta^6 \\cos(4\\theta)}{\\sin^2(4\\theta)}$$\n\nAlternatively, you can factorise out common terms (such as $6\\theta^5$) from the numerator:\n\n$$\\frac{ds}{d\\theta} = \\frac{6\\theta^5 [3\\sin(4\\theta) - 2\\theta\\cos(4\\theta)]}{\\sin^2(4\\theta)}$$\n\nEither form is fully acceptable and earns the final accuracy mark (**A1**)."}]},{"id":"200ac662-5bb1-4a8b-af41-adf83f72abac:product_rule","label":"Product and Chain Rule","steps":[{"type":"text","order":0,"title":"Rewrite the function","content":"To use the product rule, we first rewrite the quotient $s = \\frac{3\\theta^6}{\\sin(4\\theta)}$ as a product by raising the denominator to the power of $-1$:\n\n$$s = 3\\theta^6 \\left(\\sin(4\\theta)\\right)^{-1}$$"},{"type":"text","order":1,"title":"Define product terms","content":"Now we identify the two functions $u$ and $v$ that make up our product $s = u \\cdot v$:\n\n$$u = 3\\theta^6 \\quad \\text{and} \\quad v = \\left(\\sin(4\\theta)\\right)^{-1}$$"},{"type":"text","order":2,"title":"Differentiate each part","content":"Next, we find the derivatives of $u$ and $v$ with respect to $\\theta$:\n\n1. For $u = 3\\theta^6$, using the power rule:\n$$\\frac{du}{d\\theta} = 18\\theta^5$$\n\n2. For $v = \\left(\\sin(4\\theta)\\right)^{-1}$, we use the chain rule:\n$$\\frac{dv}{d\\theta} = -1\\left(\\sin(4\\theta)\\right)^{-2} \\cdot 4\\cos(4\\theta) = -4\\cos(4\\theta)\\left(\\sin(4\\theta)\\right)^{-2}$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$u = \\htmlData{anchor=u-term}{3\\theta^6}, \\quad \\frac{du}{d\\theta} = \\htmlData{anchor=du-term}{18\\theta^5}$"},{"latex":"$$v = \\htmlData{anchor=v-term}{\\left(\\sin(4\\theta)\\right)^{-1}}, \\quad \\frac{dv}{d\\theta} = \\htmlData{anchor=dv-term}{-4\\cos(4\\theta)\\left(\\sin(4\\theta)\\right)^{-2}}$"},{"latex":"$$\\frac{ds}{d\\theta} = v \\frac{du}{d\\theta} + u \\frac{dv}{d\\theta}$"},{"latex":"$$\\frac{ds}{d\\theta} = \\htmlData{anchor=v-sub}{\\left(\\sin(4\\theta)\\right)^{-1}} \\cdot \\htmlData{anchor=du-sub}{18\\theta^5} + \\htmlData{anchor=u-sub}{3\\theta^6} \\cdot \\htmlData{anchor=dv-sub}{\\left(-4\\cos(4\\theta)\\left(\\sin(4\\theta)\\right)^{-2}\\right)}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"du-term"},{"color":"pink","style":"text-color","anchor":"v-term"},{"color":"purple","style":"text-color","anchor":"u-term"},{"color":"amber","style":"text-color","anchor":"dv-term"}],"connections":[{"to":"du-sub","from":"du-term","color":"sky","label":"du/dtheta","style":"solid"},{"to":"v-sub","from":"v-term","color":"pink","label":"v","style":"solid"},{"to":"u-sub","from":"u-term","color":"purple","label":"u","style":"solid"},{"to":"dv-sub","from":"dv-term","color":"amber","label":"dv/dtheta","style":"solid"}]},"order":3,"title":"Apply product rule","description":"Now we substitute these derivatives into the product rule formula: $$\\frac{ds}{d\\theta} = v \\frac{du}{d\\theta} + u \\frac{dv}{d\\theta}$$"},{"type":"text","order":4,"title":"Simplify final expression","content":"Finally, we simplify the terms and express them with a common denominator:\n\n$$\\frac{ds}{d\\theta} = \\frac{18\\theta^5}{\\sin(4\\theta)} - \\frac{12\\theta^6\\cos(4\\theta)}{\\sin^2(4\\theta)}$$\n\nBy multiplying the first term's numerator and denominator by $\\sin(4\\theta)$ to combine them over a common denominator, we get:\n\n$$\\frac{ds}{d\\theta} = \\frac{18\\theta^5\\sin(4\\theta) - 12\\theta^6\\cos(4\\theta)}{\\sin^2(4\\theta)}$$\n\nThis can also be factored as:\n\n$$\\frac{ds}{d\\theta} = \\frac{6\\theta^5\\left(3\\sin(4\\theta) - 2\\theta\\cos(4\\theta)\\right)}{\\sin^2(4\\theta)}$$"}]}],"generatedAt":"2026-06-09T09:11:28.904Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":null,"walkthroughVideo":null}],"subjectCode":"a_level_maths__ocr_mei","summaries":[{"id":"43670804-b308-4c43-b1b0-15ba09240572","difficultyBand":"medium","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"233cef29-b34e-452d-b329-eeb2f8d1af50","difficultyBand":"medium","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"d7542af0-c3ba-44ed-81cd-ab6ab6f8a30a","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"dcc84617-b6eb-4564-b8e5-ed5142a90f67","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"0720e9a2-40c4-4527-8636-f1396bc14a09","difficultyBand":"medium","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"1fe24200-387c-4d5a-aadd-796df3778a45","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"9548d2cd-09f8-45c9-8239-a49aa6d6bb6d","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"e90e6e7d-87c9-419a-b761-350ef207d2f5","difficultyBand":"easy","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"62776d45-a6ee-448a-bde0-84f38af196c4","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"78948c35-3ae0-4e85-af96-a4ab24b2a905","difficultyBand":"medium","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"fc58532d-2057-4310-9ca0-ac9832b1dbe5","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"1b1db3e5-c4d6-446a-ba82-54b917bb77f8","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"619f22d8-6582-4a8e-a8c8-fca3c4302e31","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"02a4484d-6465-47f9-8569-d80bc542a95c","difficultyBand":"hard","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"0de699ee-92d8-42c4-a6ff-32b84bb6e495","difficultyBand":"easy","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"137c633d-888b-442a-b962-213f13a99168","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"1dd6e7ed-234f-4728-8cdb-9ab46caf8736","difficultyBand":"easy","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":false},{"id":"362274ef-ce53-4c09-a816-30ac6ec89d5d","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"39e689e7-2e31-4a3a-9d70-526672dcf8d9","difficultyBand":"medium","calculatorUse":null,"totalMarks":5,"questionType":"LA","isCalculation":true},{"id":"56a69fac-7065-42e6-b30b-a21da9d2291a","difficultyBand":"medium","calculatorUse":null,"totalMarks":8,"questionType":"LA","isCalculation":true},{"id":"6b1223c3-3eb3-46d9-b189-b43dfd983e74","difficultyBand":"medium","calculatorUse":null,"totalMarks":12,"questionType":"LA","isCalculation":true},{"id":"b3657bab-3488-4ffa-af71-3222ec757ec3","difficultyBand":"medium","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true},{"id":"ccae4139-d718-4443-8d1b-9e52e56a46fe","difficultyBand":"easy","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"d5404ddf-d8b4-40da-8504-5e685b72f033","difficultyBand":"hard","calculatorUse":null,"totalMarks":15,"questionType":"LA","isCalculation":true},{"id":"d78e5e76-d516-4aec-8878-4041448d7b0c","difficultyBand":"hard","calculatorUse":null,"totalMarks":16,"questionType":"LA","isCalculation":true},{"id":"e963c301-9c90-428f-8a7e-ce3d26f8289c","difficultyBand":"hard","calculatorUse":null,"totalMarks":13,"questionType":"LA","isCalculation":true},{"id":"ec691163-01bf-44ff-9d8e-59875ab7ab0a","difficultyBand":"medium","calculatorUse":null,"totalMarks":9,"questionType":"LA","isCalculation":true},{"id":"f741e45a-3beb-4b3f-8712-14a39d812f97","difficultyBand":"hard","calculatorUse":null,"totalMarks":13,"questionType":"LA","isCalculation":true},{"id":"f7c42d77-ed0f-4ad7-8c97-750efd22a0ad","difficultyBand":"hard","calculatorUse":null,"totalMarks":7,"questionType":"LA","isCalculation":true}],"title":"1.9.13 Quotient rule (A-level only)","topicId":"2af73925-f8b7-4780-a294-32d3d331b72e","topicSidebar":"$L3a","topicSlug":"quotient-rule","topicTitle":"1.9.13 Quotient rule (A-level only)"}]}],"$L3b","$L3c"]}]

1.9.13 Quotient rule (A-level only)

Question 3

1.9.13 Quotient rule (A-level only) Questions

1.9.13 Quotient rule (A-level only)

Question 3

1.9.13 Quotient rule (A-level only) Questions