# Derivative of an inverse question

## Homework Statement

f(x)=(x^3)+x. If h(x) is the inverse of f(x), find h'(2).

## Homework Equations

(F$$^{-1}$$)'(x)=$$\frac{1}{F'(F^{-1}x)}$$

## The Attempt at a Solution

I want to find h'(2)=(F$$^{-1}$$)'(2)=$$\frac{1}{F'(F^{-1}(2))}$$

I know f'(x)=3(x^2)+1, so I just need to find h(2), but I don't know how to solve f(x)=(x^3)+x for its inverse. Is it possible to solve for x and then switch x and y with this type of function?

It might be difficult finding the inverse function, but it's easy to see that f(1)=2, so h(2)=1.

Mark44
Mentor
Since h is the inverse of f, h(f(x)) = x
Differentiating, you get h'(f(x))*f'(x) = 1, so h'(f(x)) = 1/f'(x).

Can you work in grief's comment that f(1) = 2 to find h'(2)? You will also need to find f'(x), and from that f'(1).