Question

Differentiate the following function:
${\left(\sin x\right)}^x+{\sin }^{-1}\sqrt{x}$

Answer 1

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

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

$\Rightarrow \log u=x\log \left(\sin x\right)$

$\Rightarrow \frac{1}{u}=\frac{du}{dx}=x\frac{\cos x}{\sin x}+\log \left(\sin x\right)$

$\Rightarrow \frac{du}{dx}=(x\cot x+\log (\sin x\left)\right)(\sin x{)}^x$..........(1)

$v={\sin }^{-1}\sqrt{x}$

$\Rightarrow \frac{dv}{dx}=\frac{1}{\sqrt{1-x}}\left(\frac{1}{2\sqrt{x}}\right)$

$\Rightarrow \frac{dv}{dx}=\frac{1}{2\sqrt{x(1-x)}}$........(2)

$\therefore \frac{dy}{dx}=\frac{du}{dx}+\frac{dv}{dx}$