# Derivative of Inverse

Gold Member

## Homework Statement

If f(x) = the third root of (x-8), find the derivative of its inverse.

## Homework Equations

The derivative of its inverse = 1/f'(f^-1(x)) or 1 over its derivative at its inverse.

## The Attempt at a Solution

I followed both the formula to verify my solution and also did some manual work in lieu of the formula. Basically, if g is the inverse of f, then f(g(x)) = x, and so the derivative of both sides is f'(g(x)g'(x) = 1, and since g'(x) is the derivative of the inverse of x, g'(x) = 1/f'(g(x)) and I get 3x^2. I have no idea if that's correct however. Note that g(x) = f^-1'(x); I'm typing g(x) since the latter notation gets messy really quickly when you don't bother to use LATEX. :p.

https://scontent-b-mia.xx.fbcdn.net/hphotos-prn2/v/1457675_10201005261715465_14857176_n.jpg?oh=9da6ee66e6962843dfa1fcab48d3c201&oe=527B6402

mfb
Mentor
There is an easy way to check your work: calculate g(x), calculate the derivative, compare it with your result.

Your final answer is correct but you actually don't need the inverse to find its derivative. There's a simpler way.

We have
$$g(f(x))=x$$
where g(x) is the inverse of f(x).
$$\Rightarrow g'(f(x))\cdot f'(x)=1 \Rightarrow g'(f(x))=\frac{1}{f'(x)}$$

Also,
$$f'(x)=\frac{1}{3(x-8)^{2/3}}=\frac{1}{3f^2(x)}$$
Substituting this
$$g'(f(x))=3f^2(x) \Rightarrow g'(x)=3x^2$$

HallsofIvy
Homework Helper
But in this particular case it is easier to find the inverse and then differentiate!

But in this particular case it is easier to find the inverse and then differentiate!

Yes, it is. I just wanted to show an alternative approach because it won't be easy to find the inverse always. For example, I recently encountered the following problem:

Let ##\displaystyle f(x)=\int_2^x \frac{dt}{\sqrt{1+t^4}}## and g(x) be the inverse of f(x). Find g'(0).

Do you think it would be easy or even possible to find the inverse?

Yes, it is. I just wanted to show an alternative approach because it won't be easy to find the inverse always. For example, I recently encountered the following problem:

Let ##\displaystyle f(x)=\int_2^x \frac{dt}{\sqrt{1+t^4}}## and g(x) be the inverse of f(x). Find g'(0).

Do you think it would be easy or even possible to find the inverse?

Looks like $\sqrt{17}$ to me.

##\displaystyle f(x)=\int_2^x \frac{dt}{\sqrt{1+t^4}}##

Could you find a solution to this integral?