ds","topicId":"9705d6f3-ffa4-4174-914d-d80850bcc545","topicSlug":"chain-rule","cards":[{"id":"53611ea8-fb30-44aa-af0e-12a7c5d97385","cardType":"basic","front":"What is the inner function in $y=(3x+1)^5$?","back":"$3x+1$","points":1,"mcqOptions":null},{"id":"de915245-0491-4a9f-a3c9-1ad863a1c0a2","cardType":"basic","front":"What is the outer function in $y=\\sqrt{5x-2}$ after setting $u=5x-2$?","back":"$y=u^{\\frac12}$","points":1,"mcqOptions":null},{"id":"65addd7b-f2be-45d3-97bc-920745d968b3","cardType":"basic","front":"Why does $x^2+\\sin x$ not require the chain rule?","back":"It is a sum of separate functions, not a composite.","points":2,"mcqOptions":null},{"id":"09492bd2-75cf-4293-a7d0-3f487e6840c2","cardType":"cloze","front":"If $y=f(u)$ and $u=g(x)$, then $\\text{[...]}$.","back":"If $y=f(u)$ and $u=g(x)$, then $\\boxed{\\frac{dy}{dx}=\\frac{dy}{du}\\frac{du}{dx}}$.","points":null,"mcqOptions":null},{"id":"df9ad83b-1926-4075-a243-8ca9b0f500c4","cardType":"basic","front":"When differentiating a composite function, what must multiply the outer derivative?","back":"The derivative of the inner function.","points":1,"mcqOptions":null},{"id":"533f11dc-7216-4583-b1e7-6c853a019f1a","cardType":"cloze","front":"For $y=[g(x)]^n$, the derivative is $\\text{[...]}$.","back":"For $y=[g(x)]^n$, the derivative is $\\boxed{\\frac{dy}{dx}=n[g(x)]^{n-1}g'(x)}$.","points":null,"mcqOptions":null},{"id":"32479cba-4625-4923-a1d9-12eb8ec5ebd2","cardType":"basic","front":"Differentiate $y=(4x-3)^6$.","back":"$\\frac{dy}{dx}=24(4x-3)^5$","points":2,"mcqOptions":null},{"id":"e37ae50d-8afa-4db6-a90f-5b409c3a554a","cardType":"basic","front":"What factor is missing from the incorrect derivative $6(4x-3)^5$ of $(4x-3)^6$?","back":"$4$, the inner derivative.","points":1,"mcqOptions":null},{"id":"87587e04-4045-4d2f-ab70-2bd3b33c7ddb","cardType":"basic","front":"Differentiate $y=(x^2+4)^{-3}$.","back":"$\\frac{dy}{dx}=-\\frac{6x}{(x^2+4)^4}$","points":2,"mcqOptions":null},{"id":"fc9f9631-0972-4c09-9b57-cce25134fbfb","cardType":"cloze","front":"$$\\frac{d}{dx}\\left(\\sin(g(x))\\right)=$ $\\text{[...]}$.","back":"$$\\frac{d}{dx}\\left(\\sin(g(x))\\right)=$ $\\boxed{\\cos(g(x))g'(x)}$.","points":null,"mcqOptions":null},{"id":"403d2040-7e86-46e4-b3ae-011a5ce4b3f8","cardType":"cloze","front":"$$\\frac{d}{dx}\\left(\\cos(g(x))\\right)=$ $\\text{[...]}$.","back":"$$\\frac{d}{dx}\\left(\\cos(g(x))\\right)=$ $\\boxed{-\\sin(g(x))g'(x)}$.","points":null,"mcqOptions":null},{"id":"87fb28d0-6806-4f6b-b2ce-f894c0e2703a","cardType":"cloze","front":"$$\\frac{d}{dx}\\left(e^{g(x)}\\right)=$ $\\text{[...]}$.","back":"$$\\frac{d}{dx}\\left(e^{g(x)}\\right)=$ $\\boxed{e^{g(x)}g'(x)}$.","points":null,"mcqOptions":null},{"id":"052dc21a-31fb-4989-ba2c-a88b19084922","cardType":"basic","front":"Differentiate $y=\\sin(3x^2-1)$.","back":"$\\frac{dy}{dx}=6x\\cos(3x^2-1)$","points":2,"mcqOptions":null},{"id":"8e375d6a-d1e0-4bc5-9dde-d07587a94330","cardType":"basic","front":"How many chain-rule derivative factors are needed for a function with three nested layers?","back":"One factor for each layer.","points":1,"mcqOptions":null},{"id":"c6ba62a8-7bab-4b89-852e-089c547dd96e","cardType":"basic","front":"Differentiate $y=e^{(x^2+1)^3}$.","back":"$\\frac{dy}{dx}=6x(x^2+1)^2e^{(x^2+1)^3}$","points":3,"mcqOptions":null},{"id":"afd4adf1-5923-4e23-a30a-0a1d0790de5e","cardType":"cloze","front":"If $V$ depends on $r$ and $r$ depends on $t$, then $\\text{[...]}$.","back":"If $V$ depends on $r$ and $r$ depends on $t$, then $\\boxed{\\frac{dV}{dt}=\\frac{dV}{dr}\\frac{dr}{dt}}$.","points":null,"mcqOptions":null},{"id":"d0e2c44c-484a-4710-b18c-2da626654c56","cardType":"basic","front":"Why should values be substituted after differentiating in a connected-rates problem?","back":"Early substitution can turn a variable into a constant, losing rate information.","points":2,"mcqOptions":null},{"id":"c2cc4ca1-4e76

1.7.18 Chain rule (A-level only) Flashcards

1.7.18 Chain rule (A-level only) Flashcards