Getting the derivative of the inverse of a function

Question

Given $f(x)$, how would I find $(f^{-1})'(x)$?

As an example how would I find that for this problem:

$f(x) = 4x^3 + 5x + 2$

Answer 1

HINTS:

$$ y = 4 x^3 + 5 x + 2 $$

$$ \frac{dy}{dx} = 12 x^2 + 5 $$

Inverse function

$$ X = 4 Y^3 + 5 Y + 2 $$

$$ \frac{dX}{dY} = \frac{1}{\frac{dy}{dx}} = 12 Y^2 + 5 $$

Graphs (x-y) and (X-Y) can be reflected (mirrored) along straight line $ \dfrac{y}{x}=\dfrac{Y}{X}= 1 $

The graph of cubic reveals that at places there can be 3 real values of $Y$ for a given $X$ and others only one, and two special cases with one coincident root and one real.

Answer 2

Assuming that the function is from $\mathbb{R}$ to $\mathbb{R}$, show that the function is a bijection and then if it is, follow the @Gudson Chou comment. Note that as the function is strictly increasing ,i.e, $f'(x)>0$ it is one to one.