Question

What is the derivative of `f(x)=e^(pix)*cos(6x)` ?

Answer 1

The answer is `f'(x)=pi e^(pi x)cos(6x)-6e^(pi x)sin(6x)`.

It looks a little complicated, but break it down into pieces that you know how to solve. On the highest level, we see a product of 2 functions, so you should be thinking product rule:

`g(x)=e^(pi x)`
`h(x)=cos(6x)`
`f(x)=g(x)h(x)`
and `f'=g'*h+g*h'`

Looking at both `g(x)` and `h(x)`, you should notice that both are composition of functions. This means that we need the chain rule:

`g(x)=j(k(x))`
`g'(x)=j'(k(x))*k'(x)`
`g'(x)=pi e^(pi x)`
`h'(x)=6 sin(6x)`

And we get the final answer by substituting:

`f'(x)=g'(x)h(x)+g(x)h'(x)`
`=pi e^(pi x) cos(6x)+ e^(pi x)6 sin(6x)`

and rearrange to get the answer on the first line.