Question

If `f(x) =xe^x` and `g(x) = sin3x`, what is `f'(g(x))`?

Answer 1

I think you meant to write this as: what is `(f(g(x)))'`? (or `d/dx(f(g(x)))`?...since it's supposed to be a Chain Rule problem). In that case, the answer is

`3e^{sin(3x)}cos(3x)+3e^{sin(3x)}cos(3x)sin(3x)`.

Explanation:

Let `h(x)=f(g(x))=e^{sin(3x)}sin(3x)`. We want to find `h'(x)=(f(g(x)))'=d/dx(f(g(x)))`.

The Chain Rule says that `h'(x)=f'(g(x)) * g'(x)`.

In this case, the Product Rule gives
`f'(x)=e^[x}+xe^[x}` and we also know (by the Chain Rule again), that `g'(x)=cos(3x) * d/dx(3x)=3cos(3x)`.

Putting these things together leads to the final answer:

`h'(x)=3e^{sin(3x)}cos(3x)+3e^{sin(3x)}cos(3x)sin(3x)`.