Question

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

Answer 1

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

Taking log on both sides

$logy=xlog\sin x+\sin xlogx$

Differentiating both sides we get

$\Rightarrow \frac{1}{y}\frac{dy}{dx}=x\frac{d}{dx}log\sin x+log\sin x+\sin x\frac{d}{dx}\left(logx\right)+logx\frac{d}{dx}\left(\sin x\right)=\frac{x}{\sin x}\cos x+log\sin x+\sin x.\frac{1}{x}+logx\cos x=xcotx+logsinx+\frac{\sin x}{x}+logx\cos x\therefore \frac{dy}{dx}=\left(\right(sinx{)}^x+x^{sinx}\left)\right(xcotx+logsinx+\frac{sinx}{x}+logxcosx)$