Collection of Maths Problems

Inverse functions

Task number: 3011

• 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 $y=\sin x$.

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