# Inverse Derivative Question

1. Oct 28, 2007

### Mach

1. The problem statement, all variables and given/known data

Assume the function f(x)=x^3+x has an inverse in R. Determine d/dx(f^-1(x)) at x=2.

2. Relevant equations

(f^-1(x))'=1/f'(f^-1(x))

3. The attempt at a solution

y'=3x^2+1

(f^-1(x))'=1/(3y^2+1)

now i substitute f^-1(2), y=10, into the equation but do not get the correct answer. I suspect that the mistake i am making is becuase i do not fully understand the question. Any help is greatly appreciated.

2. Oct 28, 2007

### ZioX

f^-1 is the functional inverse, not the reciprocal.

3. Oct 30, 2007

### nizi

$$\frac{dy}{dx} = 3 x^2 + 1$$
using implicit differentiaion
$$\frac{dx}{dy} = \frac{1}{\frac{dy}{dx}} = \frac{1}{3 x^2 + 1}$$
accordingly replacing x by y
$$\frac{dy}{dx} = \frac{1}{3 y^2 + 1}$$
i.e.
$$\frac{dy}{dx} = \frac{1}{3 \left( {x^3 + x } \right)^2 + 1}$$
then you have only to substituting $$x=2$$.