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Derivatives of Inverse Functions

  • Thread starter jumbogala
  • Start date
  • #1
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Homework Statement


f(x) = x3+ x. Note that f(2) = 10. Find (f-1)`(10).


Homework Equations





The Attempt at a Solution


Note that where I have written ` it denotes prime (as in the derivative of).

- Switch the x and y variables. x= y3 + y

- Differentiate implicitly 1= (3y2 + 1)y`

- Solve for y`. y`= 1 / (3y2 + 1). Since y = f-1, then y` = (f-1)`

Therefore (f-1)` = 1 / (3y2 + 1)

Since f(2) = 10, f-1(10)=2. But my equation is(f-1)`, not just
f-1!! I don't know what to do after this point!
 

Answers and Replies

  • #2
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- Solve for y`. y`= 1 / (3y2 + 1). Since y = f-1, then y` = (f-1)`

Therefore (f-1)` = 1 / (3y2 + 1)

Since f(2) = 10, f-1(10)=2. But my equation is(f-1)`, not just
f-1!! I don't know what to do after this point!
Since you let [tex]y=f^{-1}(x)[/tex], what you actually have is [tex](f^{-1})'(x) = \frac{1}{3(f^{-1}(x))^2 + 1}[/tex].
 
Last edited:
  • #3
423
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Okay, and we want x to be 10.

So (f-1)`(10)= 1/13

Okay, I think that makes sense now. Thank you!
 
  • #4
HallsofIvy
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Homework Helper
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Another way: f(x)= [itex]x^3+ x[/itex] so f'(x)= [itex]3x^2+ 1[/itex] and f'(2)= 13. Since f(2)= 10, f-1(10)= 2 and f'(10)= 1/13.
 

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