Question

Differentiate

${\left({\sin }^{-1}{x}^4\right)}^4$

Answer 1

$\mathrm{Let}y={\left({\sin }^{-1}{x}^4\right)}^4$
Differentiate it with respect to x we get,
$$\begin{gathered} \frac{\mathit{d}y}{\mathit{d}x}=\frac{d}{dx}{\left({\sin }^{-1}{x}^4\right)}^4 \\ =4{\left({\sin }^{-1}{x}^4\right)}^3\frac{d}{dx}\left({\sin }^{-1}{x}^4\right)\left[\mathrm{Using}\mathrm{chain}\mathrm{rule}\right] \\ =4{\left({\sin }^{-1}{x}^4\right)}^3\frac{1}{\sqrt{1-{\left(x^4\right)}^2}}\frac{d}{dx}\left(x^4\right)\left[\mathrm{Using}\mathrm{chain}\mathrm{rule}\right] \\ =4{\left({\sin }^{-1}{x}^4\right)}^3\frac{4x^3}{\sqrt{1-x^8}} \\ =\frac{16x^3{\left({\sin }^{-1}{x}^4\right)}^3}{\sqrt{1-x^8}} \\ \mathrm{So},\frac{d}{dx}{\left({\sin }^{-1}{x}^4\right)}^4=\frac{16x^3{\left({\sin }^{-1}{x}^4\right)}^3}{\sqrt{1-x^8}} \end{gathered}$$