Given:
[tex]y=x^4\sin ^{-1}(x^4)[/tex]
Differentiate with respect to x.
[tex]\frac{dy}{dx}=\frac{d(x^4\sin ^{-1}(x^4))}{dx}[/tex]
[tex]\text{Use (uv)'=uv'+vu', here u=x}^4\text{ and v=}\sin ^{-1}(x^4).[/tex]
[tex]\frac{dy}{dx}=\sin ^{-1}(x^4)\frac{d(x^4)}{dx}+x^4\frac{d(\sin^{-1}(x^4))}{dx}[/tex][tex]\text{ Use }\frac{dx^4}{dx}=4x^3\text{ and }\frac{d(\sin^{-1}(x^4))}{dx}=\frac{\frac{dx^4}{dx}}{\sqrt[]{1-(x^4)^2}}\text{.}[/tex]
[tex]\frac{dy}{dx}=\sin ^{-1}(x^4)\times4x^3+x^4\frac{\frac{dx^4}{dx}}{\sqrt[]{1-(x^4)^2}}[/tex]
[tex]\frac{dy}{dx}=\sin ^{-1}(x^4)\times4x^3+x^4\frac{4x^3}{\sqrt[]{1-x^8}}[/tex]
[tex]\frac{dy}{dx}=4x^3\lbrack\sin ^{-1}(x^4)+\frac{x^4}{\sqrt[]{1-x^8}}\rbrack[/tex]
Hence the first derivative of the given equation is
[tex]\frac{dy}{dx}=4x^3\lbrack\frac{x^4}{\sqrt[]{1-x^8}}+\sin ^{-1}(x^4)\rbrack[/tex]
