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

1 Answer
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.