Question

Differentiate the function with respect to x:
$(\sin x{)}^x+{\sin }^{-1}\sqrt{x}$

Answer 1

$y=(\sin x{)}^x+{\sin }^{-1}\sqrt{x}$

$lett=(\sin x{)}^x$

$logt=xlog\left(\sin x\right)$

Differentiating both sides wrt x

$\frac{1}{t}\frac{dt}{dx}=1.log\left(sinx\right)+\frac{xcosx}{sinx}$

$\frac{dt}{dx}=\left(sinx{)}^x\right(xcotx+logsinx)$

$y=t+si{n}^{-1}\sqrt{x}$

Differentiating both sides wrt x

$\frac{dy}{dx}=\frac{dt}{dx}+\frac{d\left(si{n}^{-1}\sqrt{x}\right)}{dx}$

$\frac{dy}{dx}=\frac{1}{2\sqrt{x(1-x)}}+\left(sinx{)}^x\right(xcotx+logsinx)$$