Find the derivative of inverse of this function

[utkarshakash]

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.

Homework Equations

The Attempt at a Solution

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

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

I don't know the integration ahead.

 

[Curious3141]

Homework 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.

Homework Equations

The Attempt at a Solution

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

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

 

[utkarshakash]

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]

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

Correct.