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.

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Answer 2

The derivative of f(x) = e^(pix) * cos(6x) with respect to x is -pie^(pix) * sin(6x) + 6e^(pix) * cos(6x).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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