Find the derivative of inverse of this function

  • #1
utkarshakash
Gold Member
855
13

Homework Statement


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

Homework Equations



The Attempt at a Solution


[itex]\frac{dy}{dx}=y+e^x \sqrt{x^4+1} \\
dy=(y+e^x \sqrt{x^4+1})dx[/itex]

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

I don't know the integration ahead.
 

Answers and Replies

  • #2
Curious3141
Homework Helper
2,850
87

Homework Statement


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

Homework Equations



The Attempt at a Solution


[itex]\frac{dy}{dx}=y+e^x \sqrt{x^4+1} \\
dy=(y+e^x \sqrt{x^4+1})dx[/itex]

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

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
 
  • #3
utkarshakash
Gold Member
855
13
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?
 
  • #4
Curious3141
Homework Helper
2,850
87
OK. I followed your method and got the answer as 1/3. Is this correct?

Correct.
 

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