Which of the following statements are true? I. d/dx(uv) = du/dx · dv/dx. II. d/dx(e^2x) = 2e^2x. III. d/dx(log x) = 1/x for x > 0. IV. d/dx(cos x) = sin x.

Derivatives & Second-Order Derivatives · Question 12 of 96

Question 12

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Which of the following statements are true?
I. $\frac{d}{dx}(uv) = \frac{du}{dx} \cdot \frac{dv}{dx}$ .
II. $\frac{d}{dx}\left(e^{2x}\right) = 2e^{2x}$ .
III. $\frac{d}{dx}(\log x) = \frac{1}{x}$ for $x > 0$ .
IV. $\frac{d}{dx}(\cos x) = \sin x$ .

II and III onlyCorrect answerYour answerI, II and III onlyCorrect answerYour answerII, III and IV onlyCorrect answerYour answerIII onlyCorrect answerYour answer

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CorrectNot quiteAnswer A

Correct answer: A

II is the chain rule on $e^{2x}$ , and III is a standard derivative, so both are true.
I is false: $(uv)' = u'v + uv'$ . IV is false: $\frac{d}{dx}(\cos x) = -\sin x$ .

Concept: The rules of differentiation
Typical timeTypical: 120 s
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The rule this testsThe rules of differentiation

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