Inverse functions

Differentiate the following functions, using the formula for the derivative of an inverse function

Variant 1
\( \sqrt[n]{x} \)
Hint
For \(y=f(x)\) use the relationship \(\displaystyle f'(x)=\frac1{(f^{-1}(y))'}\).
Resolution
Let \(y=\sqrt[n]{x}\).

\(\displaystyle (\sqrt[n]{x})'= \frac{1}{(y^n)'}= \frac{1}{n y^{n-1}}= \frac{1}{n \left(\sqrt[n]{x}\right)^{n-1}}= \frac{1}n{x^{\frac1n-1}} \)
Variant 2
\( \arcsin{x} \)
Resolution
Let \(y=\sin x\).

\(\displaystyle (\arcsin x)'= \frac{1}{(\sin y)'}= \frac{1}{\cos y}= \frac{1}{\sqrt{1-\sin^2 y}}= \frac{1}{\sqrt{1-x^2}} \)