Question

$y=2x^{\sin x}-({\sin }^{-1}x{)}^2$. find $\frac{dy}{dx}$

Answer 1

Consider the following differential eq.

$y=2x^{\sin x}-{\left(si{n}^{-1}x\right)}^2$

$y=u-v$

Differentiate w.r.t $x$

$\frac{dy}{dx}=\frac{du}{dx}-\frac{dv}{dx}..........\left(1\right)$

let , $u$ $v$ and differntiate w.r.t $x$

$u=x^{\sin x}$

$\log u=x\sin x$

$\frac{1}{u}\frac{du}{dx}=\cos x\log x+\frac{\sin x}{x}$

$\frac{du}{dx}=\frac{(\cos x\log x+\sin x)}{x}{x}^{\sin x}.....\left(2\right)$

$\frac{dv}{dy}=2{\left({\sin }^{-1}x\right)}^{}\frac{1}{\sqrt{1-x^2}}........\left(3\right)$

$\frac{dy}{dx}=\frac{(\cos x\log x+\sin x)}{x}{x}^{\sin x}-2{\left({\sin }^{-1}x\right)}^{}\frac{1}{\sqrt{1-x^2}}$

Hence, this is the required answer