# How do you find the derivative of an inverse function of f(x)=x^5+3x−2?

May 11, 2018

$\frac{d}{\mathrm{dx}} \left({f}^{- 1} \left(x\right)\right) = \frac{1}{5 {\left({f}^{- 1} \left(x\right)\right)}^{4} + 3}$

#### Explanation:

The function $f \left(x\right)$ is described by the equation:

$y = {x}^{5} + 3 x - 2$

An equation describing its inverse can be formed by swapping $x$ and $y$ to get:

$x = {y}^{5} + 3 y - 2$

Then taking the derivative with respect to $x$ we find:

$1 = 5 {y}^{4} \frac{\mathrm{dy}}{\mathrm{dx}} + 3 \frac{\mathrm{dy}}{\mathrm{dx}}$

So:

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{1}{5 {y}^{4} + 3}$

That is:

$\frac{d}{\mathrm{dx}} \left({f}^{- 1} \left(x\right)\right) = \frac{1}{5 {\left({f}^{- 1} \left(x\right)\right)}^{4} + 3}$