":"The standard trigonometric derivatives are $$\\frac{d}{dx}(\\sin x)=\\text{\\text{[...]}}$$ and $$\\frac{d}{dx}(\\cos x)=\\text{-\\sin x}.$$","back":"The standard trigonometric derivatives are $$\\frac{d}{dx}(\\sin x)=\\text{\\boxed{\\cos x}}$$ and $$\\frac{d}{dx}(\\cos x)=\\text{-\\sin x}.$$","points":null,"mcqOptions":null},{"id":"1870a368-a502-4973-acfc-ce796d18cc5b","cardType":"cloze","front":"The standard trigonometric derivatives are $$\\frac{d}{dx}(\\sin x)=\\text{\\cos x}$$ and $$\\frac{d}{dx}(\\cos x)=\\text{\\text{[...]}}.$$","back":"The standard trigonometric derivatives are $$\\frac{d}{dx}(\\sin x)=\\text{\\cos x}$$ and $$\\frac{d}{dx}(\\cos x)=\\text{\\boxed{-\\sin x}}.$$","points":null,"mcqOptions":null},{"id":"dd856d72-3da8-46b3-90ad-1fca2f2fb3ef","cardType":"basic","front":"What is $\\frac{d}{dx}(e^x)$?","back":"$e^x$","points":1,"mcqOptions":null},{"id":"2fa0d5a0-43af-434a-ba39-56506eabfe8e","cardType":"basic","front":"What is $\\frac{d}{dx}(\\ln x)$?","back":"$\\frac{1}{x}$","points":1,"mcqOptions":null},{"id":"0b60ca24-3b8e-412d-b780-4d6e04bda7cb","cardType":"cloze","front":"If $y=uv$, the product rule gives $$\\text{[...]}$$.","back":"If $y=uv$, the product rule gives $$\\boxed{\\frac{dy}{dx}=u\\frac{dv}{dx}+v\\frac{du}{dx}}$$.","points":null,"mcqOptions":null},{"id":"0c195616-4bd2-4248-97ab-398b4ba29975","cardType":"basic","front":"How does the product rule differentiate two factors?","back":"Differentiate each factor once, leaving the other unchanged, then add.","points":2,"mcqOptions":null},{"id":"97125460-53e5-4205-8769-c5c7704f9cd5","cardType":"basic","front":"Why is $(uv)'=u'v'$ generally incorrect?","back":"The product rule requires two terms joined by addition.","points":2,"mcqOptions":null},{"id":"5e8f7497-4226-4f7f-a65e-c43a4bae272e","cardType":"cloze","front":"If $y=\\frac{u}{v}$, the quotient rule gives $$\\text{[...]}$$.","back":"If $y=\\frac{u}{v}$, the quotient rule gives $$\\boxed{\\frac{dy}{dx}=\\frac{v\\frac{du}{dx}-u\\frac{dv}{dx}}{v^2}}$$.","points":null,"mcqOptions":null},{"id":"5eb8afbe-9072-43b9-b915-98db6c3dbee6","cardType":"basic","front":"Where are $\\frac{x^2+1}{x-2}$ and its derivative undefined?","back":"$x=2$, because the denominator is zero.","points":1,"mcqOptions":null},{"id":"4d2086e0-0ecb-4e4e-8662-8ee9bddcc66c","cardType":"cloze","front":"The chain rule gives $$\\text{[...]}$$, while $$\\frac{d}{dx}([f(x)]^n)=\\text{n[f(x)]^{n-1}f'(x)}.$$","back":"The chain rule gives $$\\boxed{\\frac{dy}{dx}=\\frac{dy}{du}\\frac{du}{dx}}$$, while $$\\frac{d}{dx}([f(x)]^n)=\\text{n[f(x)]^{n-1}f'(x)}.$$","points":null,"mcqOptions":null},{"id":"a45a881f-2b43-4fa4-acfe-a04981934189","cardType":"cloze","front":"The chain rule gives $$\\frac{dy}{dx}=\\frac{dy}{du}\\frac{du}{dx}$$, while $$\\frac{d}{dx}([f(x)]^n)=\\text{\\text{[...]}}.$$","back":"The chain rule gives $$\\frac{dy}{dx}=\\frac{dy}{du}\\frac{du}{dx}$$, while $$\\frac{d}{dx}([f(x)]^n)=\\text{\\boxed{n[f(x)]^{n-1}f'(x)}}.$$","points":null,"mcqOptions":null},{"id":"f5744532-e6f7-4dcd-82de-16aebd83724b","cardType":"basic","front":"What is $\\frac{d}{dx}(\\sin(f(x)))$?","back":"$\\cos(f(x))f'(x)$","points":1,"mcqOptions":null},{"id":"4f521be8-6691-4cea-8872-28e610ee9513","cardType":"basic","front":"What is $\\frac{d}{dx}(\\cos(f(x)))$?","back":"$-\\sin(f(x))f'(x)$","points":1,"mcqOptions":null},{"id":"bebf7b91-e68d-4f74-8312-c6ed0d871344","cardType":"basic","front":"What is $\\frac{d}{dx}(e^{f(x)})$?","back":"$e^{f(x)}f'(x)$","points":1,"mcqOptions":null},{"id":"c68ed36e-b016-40ef-9b23-67a13e174009","cardType":"basic","front":"What is $\\frac{d}{dx}(\\ln(f(x)))$?","back":"$\\frac{f'(x)}{f(x)}$","points":1,"mcqOptions":null},{"id":"6cf118f8-04e3-4700-9219-b50e26ce642a","cardType":"basic","front":"Which operation should determine the first differentiation rule used?","back":"The operation holding the whole expression together.","points":1,"mcqOptions":null},{"id"

3.5.2 Differentiation (A-level only) Flashcards

3.5.2 Differentiation (A-level only) Flashcards