What is d/dx(x^sin x) for x > 0?

x^sin x (log x cos x + (sin x)/x)

x^sin x (log x sin x + (cos x)/x)

Question 87

MediumOriginal question

What is $\frac{d}{dx}\left(x^{\sin x}\right)$ for $x > 0$ ?

x^{\sin x} \log x \cos x

Correct answerYour answer

x^{\sin x - 1} \sin x

Correct answerYour answer

x^{\sin x} \left(\log x \sin x + \frac{\cos x}{x}\right)

Correct answerYour answer

x^{\sin x} \left(\log x \cos x + \frac{\sin x}{x}\right)

Correct answerYour answer

Show the answer and explanation

CorrectNot quiteAnswer D

Correct answer: D

$\log y = \sin x\log x$ , so $\frac{1}{y}\frac{dy}{dx} = \cos x\log x + \frac{\sin x}{x}$ .
$\frac{dy}{dx} = x^{\sin x}\left(\cos x\log x + \frac{\sin x}{x}\right)$ .

Worth noting

$x^{\sin x}\cos x\log x$ keeps only one term of the product rule on $\sin x\log x$ .

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