Question

Find the differential coefficient of $x^{\sin x}$.

Answer 1

Let $y=x^{\sin x}$

Taking $\log$ on both the sides $\log y=\sin x\log x$

Differentiating both the sides w.r.t. $x$, $\frac{d}{dx}\log y=\frac{d}{dx}\left(\sin x\log x\right)$

$\Rightarrow \frac{1}{y}\frac{dy}{dx}=\sin x.\frac{1}{x}+\log x.\cos x$

$\Rightarrow \frac{dy}{dx}=y\left(\frac{\sin x+x\log x.\cos x}{x}\right)$

$\Rightarrow \frac{dy}{dx}=x^{\sin x}\left(\frac{\sin x+x\log x.\cos x}{x}\right)$.