}{\\color{@sky}{x+3}})(\\htmlData{anchor=t2}{\\color{@purple}{x+1}})^3$"},{"latex":"$$y = (\\htmlData{anchor=u1}{\\color{@sky}{u+2}})(\\htmlData{anchor=u2}{\\color{@purple}{u}})^3$"},{"latex":"$$y = u^4 + 2u^3$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"t1"},{"color":"purple","style":"text-color","anchor":"t2"},{"color":"sky","style":"text-color","anchor":"u1"},{"color":"purple","style":"text-color","anchor":"u2"}],"connections":[{"to":"u1","from":"t1","color":"sky","label":null,"style":"solid"},{"to":"u2","from":"t2","color":"purple","label":null,"style":"solid"}]},"order":1,"title":"Rewrite in terms of u","description":"We replace $x+3$ and $x+1$ with their expressions in terms of $u$, and then expand the brackets to make differentiation straightforward."},{"type":"text","order":2,"title":"Differentiate with respect to u","content":"Now we differentiate $y = u^4 + 2u^3$ with respect to $u$ using the power rule:\n\n$$\\frac{dy}{du} = 4u^3 + 6u^2$$\n\nSince $u = x+1$, we also have:\n\n$$\\frac{du}{dx} = 1$$\n\nBy the chain rule:\n\n$$\\frac{dy}{dx} = \\frac{dy}{du} \\cdot \\frac{du}{dx} = (4u^3 + 6u^2) \\cdot 1 = 4u^3 + 6u^2$$"},{"type":"text","order":3,"title":"Substitute back to x","content":"Now we replace $u$ with $(x+1)$ to write our derivative in terms of $x$:\n\n$$\\frac{dy}{dx} = 4(x+1)^3 + 6(x+1)^2$$"},{"type":"text","order":4,"title":"Factorise and simplify","content":"To get our final answer in the simplest factored form, we take out the highest common factor, which is $2(x+1)^2$:\n\n$$\\begin{aligned} \\frac{dy}{dx} &= 2(x+1)^2 \\left[ 2(x+1) + 3 \\right] \\\\ &= 2(x+1)^2 \\left[ 2x + 2 + 3 \\right] \\\\ &= 2(x+1)^2(2x+5) \\end{aligned}$$\n\n**Markscheme Notes:**\n* Identifying the derivative of $(x+1)^3$ as $3(x+1)^2$ or equivalent expansion steps earns the first method mark M1.\n* Correctly applying the rules of differentiation to get the full unsimplified derivative earns the second method mark M1.\n* Simplifying fully to $2(x+1)^2(2x+5)$ earns the final accuracy mark A1."}]}],"generatedAt":"2026-06-09T09:15:33.512Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"options":[]},{"id":"9894604a-afd0-4f36-9dd2-3a837f7d2bbb","label":"b","content":"$$5 + x^2 \\ln(3x+1)$","markscheme":"- Differentiates $x^2$ to $2x$ and $\\ln(3x+1)$ to $\\frac{3}{3x+1}$ M1\n- Applies product rule to the $x^2 \\ln(3x+1)$ term M1\n- Correct final expression $2x \\ln(3x+1) + \\frac{3x^2}{3x+1}$ A1","marks":3,"order":1,"markingRubric":{"totalMarks":3,"points":[{"id":"p1","marks":1,"description":"Correct differentiation of both components of the product term.","code":"M"},{"id":"p2","marks":1,"description":"Correct application of product rule to the $x^2 \\ln(3x+1)$ term.","code":"M"},{"id":"p3","marks":1,"description":"Correct final derivative expression.","code":"A"}],"finalAnswers":["$$2x \\ln(3x+1) + \\frac{3x^2}{3x+1}$"],"generatedAt":"2026-05-06T14:04:02.995Z","modelVersion":"gemini-3-flash-preview"},"explanation":{"methods":[{"id":"standard_differentiation","label":"Direct Differentiation","steps":[{"type":"text","order":0,"title":"Analyze the expression","content":"We want to differentiate the expression with respect to $x$:\n\n$$y = 5 + x^2 \\ln(3x+1)$$\n\nWe can differentiate this expression term-by-term:\n\n* The first term is a constant: $5$.\n* The second term is a product of two functions: $x^2$ and $\\ln(3x+1)$."},{"type":"text","order":1,"title":"Differentiate components","content":"To prepare for the product rule on the second term, we first find the derivatives of its individual components:\n\n* For $u = x^2$, the derivative is:\n $$u' = 2x$$\n\n* For $v = \\ln(3x+1)$, we apply the chain rule by differentiating the outer natural logarithm and multiplying by the derivative of the inner function, $3x+1$:\n $$v' = \\frac{1}{3x+1} \\cdot 3 = \\frac{3}{3x+1}$$\n\nThis earns the first method mark **(M1)**."},{"type":"text","order":2,"title":"Apply the product rule","content":"Recall the product rul

9.4 The Product Rule Questions

9.4 The Product Rule Questions

9.4 The Product Rule Questions

9.4 The Product Rule Questions