Question

If `g(5) = -3, g'(5) = 6, h(5) = 3, h'(5) = -2` what is `f'(5)`, if `f(x)=g(h(x))`. Is it possible to find it?

Answer 1

The trick here is to use the chain rule. In words, the chain rule states that the derivative of a composite function like `g(h(x))` is equal to the derivative of the outside function with the inside function inside it, all multiplied by the derivative of the inside function.

Expressed mathematically, if

`f(x)=g(h(x))`

then

`f'(x)=g'(h(x))*h'(x)`

Thus,

`f'(5)=g'(h(5))*h'(5)`

We know that `h(5)=3` and `h'(5)=-2`, so

`f'(5)=g'(3)*(-2)`

However, we do not know the value of `g'(3)` so it is not possible to find `f'(5)` with the given information.