Question

Find $\frac{dy}{dx}$, if $y=x^{\sin x}$.

Answer 1

$y=x^{\sin x}$

$\log y=\sin x\log x$

$\frac{d}{dx}\left(\log y\right)=\frac{d}{dx}\left(\sin x\log x\right)$

$\frac{1}{y}\frac{dy}{dx}=\sin x\times \frac{d}{dx}\left(\log x\right)+\log x\frac{d}{dx}\left(\sin x\right)$

$\frac{1}{y}\frac{dy}{dx}=\sin x\times \frac{1}{x}+\log x\sin x$

$\frac{dy}{dx}=y(\frac{\sin x}{x}+\log x\sin x)$

$\therefore \frac{dy}{dx}=x^{\sin x}(\frac{\sin x}{x}+\log x\sin x)$