For x > 0, which of the following statements are true? I. d/dx e^log(x^2) = 2x. II. d/dxlog(e^sin x) = e^sin xcos x. III. d/dx 2^log x = (2^log xlog 2)/x. IV. d/dx e^√(x) = (e^√(x))/(√(x)).

Question 80

HardOriginal question

For $x > 0$ , which of the following statements are true?
I. $\frac{d}{dx} e^{\log(x^{2})} = 2x$ .
II. $\frac{d}{dx}\log\left(e^{\sin x}\right) = e^{\sin x}\cos x$ .
III. $\frac{d}{dx} 2^{\log x} = \frac{2^{\log x}\log 2}{x}$ .
IV. $\frac{d}{dx} e^{\sqrt{x}} = \frac{e^{\sqrt{x}}}{\sqrt{x}}$ .

I, II and III onlyCorrect answerYour answerI and III onlyCorrect answerYour answerI onlyCorrect answerYour answerI, III and IV onlyCorrect answerYour answer

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CorrectNot quiteAnswer B

Correct answer: B

I: $e^{\log x^{2}} = x^{2}$ , whose derivative is $2x$ . III: $2^{u}\log 2 \cdot u'$ with $u = \log x$ , $u' = \frac{1}{x}$ .
II is false: $\log e^{\sin x} = \sin x$ , so the derivative is $\cos x$ . IV is false: $e^{\sqrt{x}} \cdot \frac{1}{2\sqrt{x}}$ has a 2 below.

Worth noting

II differentiates $e^{\sin x}$ and forgets the logarithm outside it, which undoes the exponential.

Concept: Exponential and logarithm inside a chain
Typical timeTypical: 150 s
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The rule this testsExponential and logarithmic functions

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