Question

Let `f` and `g` be functions that are differentiable everywhere. If `g` is the inverse function of `f` and if `g(-2) = 5` and `f'(5) = -1/2`, then what is `g'(-2)`?

Answer 1

`g'(-2) = -2`

Explanation:

If `g(x)` is the inverse function of `f(x)`, it follows that:

`f(g(x)) = x`

Differentiate the identity using the chain rule:

`d/dx f(g(x)) = d/dx x`

`f'(g(x)) g'(x) = 1`

So:

`g'(x) = 1/(f'(g(x))`

For `x= -2` we have:

`g'(-2) = 1/(f'(g(-2))) = 1/(f'(5)) = 1/(-1/2) = -2`