vel only) Flashcards","topicId":"1f094367-bd42-44b4-8b8f-823b71aa54df","topicSlug":"differentiation-14","cards":[{"id":"bae10f87-2fba-4dff-a5a4-37cbe363dbfb","cardType":"cloze","front":"When $y$ depends on $x$, $$\\frac{d}{dx}(y^n)=\\htmlClass{remove-space}{\\text{[...]}}.$$","back":"When $y$ depends on $x$, $$\\frac{d}{dx}(y^n)=\\htmlClass{remove-space}{\\boxed{ny^{n-1}\\frac{dy}{dx}}}.$$","points":null,"mcqOptions":null},{"id":"7ba6ee13-6442-446e-962c-b652f91a663e","cardType":"basic","front":"Differentiate $5y^4$ with respect to $x$.","back":"$$\\displaystyle \\mathbf{20y^3\\frac{dy}{dx}}$","points":1,"mcqOptions":null},{"id":"cc37f797-15ff-43e4-afa1-ddc5dd603766","cardType":"basic","front":"Why does $\\frac{dy}{dx}$ appear when differentiating a power of $y$ with respect to $x$?","back":"Because $y$ is a function of $x$, so the chain rule applies.","points":2,"mcqOptions":null},{"id":"eaee03c9-a79e-49b5-b1ce-07383926caad","cardType":"cloze","front":"In implicit differentiation, treat $y$ as [ ] before differentiating both sides with respect to $x$.","back":"In implicit differentiation, treat $y$ as a function of $x$ before differentiating both sides with respect to $x$.","points":null,"mcqOptions":null},{"id":"44c8847b-55b5-4c86-8db4-d5d1b0533aa8","cardType":"cloze","front":"Using the product rule, $\\displaystyle \\frac{d}{dx}(xy)=\\htmlClass{remove-space}{\\text{[...]}} $.","back":"Using the product rule, $\\displaystyle \\frac{d}{dx}(xy)=\\htmlClass{remove-space}{\\boxed{x\\frac{dy}{dx}+y}} $.","points":null,"mcqOptions":null},{"id":"c0daa94f-f27a-4052-b84e-49fbf46f0fd9","cardType":"cloze","front":"The line through $(x_1,y_1)$ with gradient $m$ has equation $\\text{[...]}$.","back":"The line through $(x_1,y_1)$ with gradient $m$ has equation $\\boxed{y-y_1=m(x-x_1)}$.","points":null,"mcqOptions":null},{"id":"bee76ec1-4dcf-49f6-94c3-49435c185abe","cardType":"basic","front":"If a tangent has non-zero gradient $m$, what is the normal gradient?","back":"$$\\displaystyle \\mathbf{-\\frac{1}{m}}$","points":1,"mcqOptions":null},{"id":"d12946a7-5d06-4555-9e7a-4166174971a7","cardType":"cloze","front":"For $x=f(t)$ and $y=g(t)$, $$\\frac{dy}{dx}=\\htmlClass{remove-space}{\\text{[...]}}.$$","back":"For $x=f(t)$ and $y=g(t)$, $$\\frac{dy}{dx}=\\htmlClass{remove-space}{\\boxed{\\frac{dy/dt}{dx/dt}}}.$$","points":null,"mcqOptions":null},{"id":"70852514-ca87-4b4a-be95-2a2ca0b66a57","cardType":"basic","front":"What condition is required to use $\\frac{dy}{dx}=\\frac{dy/dt}{dx/dt}$ directly?","back":"$$\\displaystyle \\mathbf{\\frac{dx}{dt}\\ne0}$","points":1,"mcqOptions":null},{"id":"adc95dbd-4e66-4146-9ce8-ec9341316bef","cardType":"basic","front":"Predict the tangent when $\\frac{dx}{dt}=0$ but $\\frac{dy}{dt}\\ne0$.","back":"The curve usually has a vertical tangent.","points":1,"mcqOptions":null},{"id":"c47a8843-2851-49cb-a47b-98778d8a16e0","cardType":"basic","front":"What can be concluded from the quotient formula if $\\frac{dx}{dt}=0$ and $\\frac{dy}{dt}=0$?","back":"No gradient can be concluded without further analysis.","points":1,"mcqOptions":null},{"id":"6d00c037-0c4f-43a7-a39c-b948038a2d43","cardType":"basic","front":"Find $\\frac{dy}{dx}$ for $x^2+y^2=25$.","back":"$$\\displaystyle \\mathbf{\\frac{dy}{dx}=-\\frac{x}{y}}$","points":2,"mcqOptions":null},{"id":"d8ce8087-2567-4650-b834-567e36565351","cardType":"basic","front":"Find the gradient of $x^2+y^2=25$ at $(3,4)$.","back":"$$\\displaystyle \\mathbf{-\\frac{3}{4}}$","points":1,"mcqOptions":null},{"id":"fe2338ef-c263-4b60-8b3a-8c61cfed12d0","cardType":"basic","front":"Find $\\frac{dy}{dx}$ for $x^2+xy+y^2=7$.","back":"$$\\displaystyle \\mathbf{\\frac{dy}{dx}=-\\frac{2x+y}{x+2y}}$","points":2,"mcqOptions":null},{"id":"98bcc638-8b4d-4120-823c-a230a352f7b2","cardType":"basic","front":"Find the tangent to $x^2+xy+y^2=7$ at $(1,2)$.","back":"$$\\displaystyle \\mathbf{y-2=-\\frac45(x-1)}$","points":2,"mcqOptions":nul

3.6.6 Differentiation (A-level only) Flashcards

3.6.6 Differentiation (A-level only) Flashcards