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

1 Answer
Amory W.
Sep 7, 2014

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.

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