# Thread: Derivative of the inverse

1. ## Derivative of the inverse

f(x) = x^3 + x

h(x) is the inverse of f(x)

Find the derivative of h(2), step by step please.

2. ## Re: Derivative of the inverse

$\displaystyle f(x)=2\Leftrightarrow x^3+x=2\Leftrightarrow x=1$ (only one real root). Now, using a well known theorem:

$\displaystyle h'(2)=\frac{1}{f'(1)}=\ldots=\frac{1}{4}$

3. ## Re: Derivative of the inverse

Originally Posted by FernandoRevilla
$\displaystyle f(x)=2\Leftrightarrow x^3+x=2\Leftrightarrow x=1$ (only one real root). Now, using a well known theorem:

$\displaystyle h'(2)=\frac{1}{f'(1)}=\ldots=\frac{1}{4}$
Thank you, but I'm not sure I understand yet. Why did you set f(x) = 2 for the first step? What process are you using?

4. ## Re: Derivative of the inverse

Originally Posted by TWN
Thank you, but I'm not sure I understand yet. Why did you set f(x) = 2 for the first step? What process are you using?
He is using the inverse function theorem,

$\displaystyle \left(f^{-1}\right)'(x) = \frac1{f'\left(f^{-1}(x)\right)}$,

which is indeed very well known. You should familiarize yourself with it.

To find $\displaystyle \left(f^{-1}\right)'(2)$, we want to know $\displaystyle f^{-1}(2)$. So we set $\displaystyle f(x) = 2\Rightarrow x=1$ so $\displaystyle f(1)=2\Rightarrow f^{-1}(2) = 1$. Now make the substitutions, and finish it.