# Find the derivative of inverse of this function

Gold Member

## Homework Statement

Let f be a real valued differentiable function defined in (-1,1). If f(0)=2 and $f'(x)=f(x)+e^x(\sqrt{x^4+1})$ , then find $\frac{df^{-1}(x)}{dx}$ at x=2.

## The Attempt at a Solution

$\frac{dy}{dx}=y+e^x \sqrt{x^4+1} \\ dy=(y+e^x \sqrt{x^4+1})dx$

Integrating both sides
$y=xy+\int (e^x\sqrt{x^4+1})dx$

I don't know the integration ahead.

Curious3141
Homework Helper

## Homework Statement

Let f be a real valued differentiable function defined in (-1,1). If f(0)=2 and $f'(x)=f(x)+e^x(\sqrt{x^4+1})$ , then find $\frac{df^{-1}(x)}{dx}$ at x=2.

## The Attempt at a Solution

$\frac{dy}{dx}=y+e^x \sqrt{x^4+1} \\ dy=(y+e^x \sqrt{x^4+1})dx$

Integrating both sides
$y=xy+\int (e^x\sqrt{x^4+1})dx$

I don't know the integration ahead.

You can't separate variables like that. Your solution of the d.e. is wrong.

But in any case, you don't need to solve that d.e. Do you know how to find the derivative of an inverse function at a given point without actually finding the inverse explicitly?

You might want to take a look at an earlier post of mine: https://www.physicsforums.com/showpost.php?p=4296589&postcount=9

Gold Member
You can't separate variables like that. Your solution of the d.e. is wrong.

But in any case, you don't need to solve that d.e. Do you know how to find the derivative of an inverse function at a given point without actually finding the inverse explicitly?

You might want to take a look at an earlier post of mine: https://www.physicsforums.com/showpost.php?p=4296589&postcount=9
OK. I followed your method and got the answer as 1/3. Is this correct?

Curious3141
Homework Helper
OK. I followed your method and got the answer as 1/3. Is this correct?

Correct.