Question

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

Answer 1

$D=\frac{d}{dx}((\frac{(sinx{)}^x}{\left(A\right)})+(\frac{\left(si{n}^{-1}\sqrt{x}\right)}{\left(B\right)}))$

$A\Rightarrow \frac{d}{dx}\left(e^{x}ln\right(sinx\left)\right)$

$\Rightarrow e^{xln\left(sinx\right)}\left(ln\right(sinx)+xcotx)$

$\Rightarrow \left(sinx{)}^x\right(ln\left(sinx\right)+xcotx)$

$B\Rightarrow \frac{d}{dx}\left(si{n}^{-1}\sqrt{x}\right)$

$=\frac{1}{\sqrt{1-(\sqrt{x}{)}^2}}.\frac{1}{2\sqrt{x}}$

$=\frac{1}{\sqrt{1-x}}.\frac{1}{2\sqrt{x}}$

$\therefore D=si{n}^x\left(x\right)\left(ln\right(sinx)+xcotx)+\frac{1}{2\sqrt{x}}\frac{1}{\sqrt{1-x}}$