Question

Differentiate $x^{{\sin }^{-1}x}$ w.r.t. ${\sin }^{-1}x.$

• A. $x^{{\sin }^{-1}x}[\log x+{\sin }^{-1}x.\frac{\sqrt{(1-x^2)}}{x}]$
• B. $-x^{{\sin }^{-1}x}[\log x+{\sin }^{-1}x\frac{\sqrt{(1-x^2)}}{x}]$
• C. $x^{{\sin }^{-1}x}[\log x+{\sin }^{-1}x\frac{\sqrt{(1+x^2)}}{x}]$
• D. $-x^{{\sin }^{-1}x}[\log x+{\sin }^{-1}x\frac{\sqrt{(1+x^2)}}{x}]$

Answer 1

The correct option is A $x^{{\sin }^{-1}x}[\log x+{\sin }^{-1}x.\frac{\sqrt{(1-x^2)}}{x}]$

Let $y=x^{{\sin }^{-1}x}$ and $z={\sin }^{-1}x\Rightarrow \sin z=x$
we have to find $\frac{dy}{dz}$
$\Rightarrow y=(\sin z{)}^z$ taking $\log$ both side $\log y=z\log \sin z$
Differentiating w.r.t $z$
$\frac{1}{y}\frac{dy}{dz}=\log \sin z+z\frac{\cos z}{\sin z}$
$\therefore \frac{dy}{dz}=y(\log \sin z+z\frac{\cos z}{\sin z})=x^{{\sin }^{-1}x}[\log x+{\sin }^{-1}x.\frac{\sqrt{(1-x^2)}}{x}]$