# Homework Help: Derivatives of Inverse Functions

1. Jan 18, 2009

### jumbogala

1. The problem statement, all variables and given/known data
f(x) = x3+ x. Note that f(2) = 10. Find (f-1)(10).

2. Relevant equations

3. 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!

2. Jan 18, 2009

### Unco

Since you let $$y=f^{-1}(x)$$, what you actually have is $$(f^{-1})'(x) = \frac{1}{3(f^{-1}(x))^2 + 1}$$.

Last edited: Jan 18, 2009
3. Jan 18, 2009

### jumbogala

Okay, and we want x to be 10.

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

Okay, I think that makes sense now. Thank you!

4. Jan 19, 2009

### HallsofIvy

Another way: f(x)= $x^3+ x$ so f'(x)= $3x^2+ 1$ and f'(2)= 13. Since f(2)= 10, f-1(10)= 2 and f'(10)= 1/13.