# Find the derivative of the inverse function

• ludi_srbin
In summary, the conversation is about finding the derivative of the inverse function of a one-to-one and continuous function f, given that f'(x)=1+[f(x)]^2. The speaker found the inverse by writing the equation f(x)=x+(1/3)[f(x)]^3 and switching the variables to get the inverse function as (1/3)x^3 - x. However, the other person disagrees with this method and suggests considering the ranges of the variables involved.
ludi_srbin
So I need to find the derivative of the inverse function. I know that f is one-to-one and is continous. Also I know that f'(x)=1+[f(x)]^2. I found the inverse writing my equation like f(x)=x+(1/3)[f(x)]^3 then I switch the variables and get that my inverse function=(1/3)x^3 - x. Then I just take the derivative and end up with x^2 - 1. Is this correct?

ludi_srbin said:
So I need to find the derivative of the inverse function. I know that f is one-to-one and is continous. Also I know that f'(x)=1+[f(x)]^2. I found the inverse writing my equation like f(x)=x+(1/3)[f(x)]^3 then I switch the variables and get that my inverse function=(1/3)x^3 - x. Then I just take the derivative and end up with x^2 - 1. Is this correct?

First, sorry for my poor english.
I don't think you are right.Because when you switch the variables in the equation you have to pay attention to the ranges of all variables.You can see in your equation f(x)=x^2-1,the range of x is(-∞,∞)，but the range of f(x) is [-1,∞).So what you do is not only switch the variables but also write the range of x at the end of your equation.

Last edited by a moderator:
Hmmm...But isn't the range of f(x) (-infinity, infinity)?

## What is the inverse function?

The inverse function is a mathematical operation that undoes another function. It essentially reverses the input and output of the original function.

## Why is finding the derivative of the inverse function important?

Finding the derivative of the inverse function is important because it allows us to understand the rate of change of the inverse function. This can be useful in solving optimization problems and in understanding the behavior of the original function.

## How do you find the derivative of the inverse function?

To find the derivative of the inverse function, you can use the formula: f'(x) = 1 / f'(f^-1(x)). This means that you take the reciprocal of the derivative of the original function evaluated at the inverse function's input.

## What if the inverse function is not easily solvable?

If the inverse function is not easily solvable, you can use implicit differentiation to find the derivative. This involves treating the inverse function as an implicit function and using the chain rule to find the derivative.

## Can the derivative of the inverse function be negative?

Yes, the derivative of the inverse function can be negative, positive, or zero. It depends on the behavior of the original function and the input of the inverse function.

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