This right here is the embodiment of the chain rule. First, I would rewrite the function as #(cos(4theta))^2#. Using the power rule, you first subtract the exponent by 1 and multiply your function by 2, keeping the inside the same. You should come up with #2(cos(4theta)#.
Next, you take the derivative of your inside, thus using the chain rule, and multiply it by the value we got previously. The derivative of #cos(4theta)# is #-sin(4theta)#. But we are not finished. We have a double chain rule, since the inside of #-sin(4theta)# has a coefficient. Thus to finish the chain rule, take the derivative of #4theta#, which is #4#.
Putting it all together, multiply all our individual pieces, to get the answer of #2(cos(4theta)(-sin(4theta)4))#. Cleaning it up, we get the final answer of #f'(x)=-8cos(4theta)(sin(4theta))#.
