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

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

(log(sin x ) + cot x) sin^x(x )

Question 99

HardOriginal question

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

x \sin^{x - 1}{\left(x \right)} \cos x

Correct answerYour answer

\log{\left(\sin x \right)} \sin^{x}{\left(x \right)}

Correct answerYour answer

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

Correct answerYour answer

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

Correct answerYour answer

Show the answer and explanation

CorrectNot quiteAnswer D

Correct answer: D

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

Worth noting

$x(\sin x)^{x - 1}\cos x$ treats the power as a constant.

Ten questions from Chain Rule, Log Differentiation & Rate of Change, at your own pace.