Question

Suppose that F(x) = f(g(x)) where g(3)=6, g(6)=13, g'(3)=4, g'(6)=2, f(3)=11, f(6)=15, f'(3)=4, and f'(6)=8. Find F'(3).

Answer 1

The rule for deriving composite functions (known as the chain rule) is:

`(f(g(x))' = f'(g(x))\cdot g'(x)`

So, in your case, we have

`F'(3) = f'(g(3))\cdot g'(3)`

We know that
`g'(3) = 4` and
`g(3)=6`

So, the expression becomes

`F'(3) = f'(6)\cdot 4`

Finally, since
`f'(6)=8`, we have

`F'(3) = 8\cdot 4 = 32`