Question

Differentiate the function with respect to x:
$x^{\sin x}+(\sin x{)}^{\cos x}$

Answer 1

Let $y=x^{\sin x}+(\sin x{)}^{\cos x}$

Taking Log both sides

$\log y=\sin x\log x+\cos x\log \left(\sin x\right)$

Now Differentiate wrt x

$\frac{1}{y}\frac{dy}{dx}=\frac{d}{dx}\left(\sin x\log x\right)+\frac{d}{dx}\left(\cos x\log \right(\sin x\left)\right)\Rightarrow \frac{1}{y}\frac{dy}{dx}=(\sin x.\frac{1}{x}+\cos x.\log x)+(\cos x.\frac{1}{\sin x}.(\cos x)-\sin x\log \sin x)\Rightarrow \frac{dy}{dx}=[\sin x.\frac{1}{x}+\cos x.\log x+\frac{{\cos }^{2}x}{\sin x}-\sin x\log \sin x][x^{\sin x}+(\sin x{)}^{\cos x}]$