# How do you find the derivative of cos^2(4theta)?

Jun 1, 2016

$f ' \left(x\right) = - 8 \cos \left(4 \theta\right) \left(\sin \left(4 \theta\right)\right)$
This right here is the embodiment of the chain rule. First, I would rewrite the function as ${\left(\cos \left(4 \theta\right)\right)}^{2}$. 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 $\cos \left(4 \theta\right)$ is $- \sin \left(4 \theta\right)$. But we are not finished. We have a double chain rule, since the inside of $- \sin \left(4 \theta\right)$ has a coefficient. Thus to finish the chain rule, take the derivative of $4 \theta$, which is $4$.
Putting it all together, multiply all our individual pieces, to get the answer of $2 \left(\cos \left(4 \theta\right) \left(- \sin \left(4 \theta\right) 4\right)\right)$. Cleaning it up, we get the final answer of $f ' \left(x\right) = - 8 \cos \left(4 \theta\right) \left(\sin \left(4 \theta\right)\right)$.