find f'(x) for the following:
f(x)= cos^5(4x-19)
I keep getting 20cos^4(4x-19)sin, but the answer is supposed to be
-20cos^4(4x-19)sin(4x-19). I can't seem to get there with derivatives
never mind. I figured it out. That is the correct anser, I didn't carry through with the derivitives.
To find the derivative of the function f(x) = cos^5(4x-19), you can use the chain rule. The chain rule states that if you have a composition of functions, such as f(g(x)), then the derivative is found by taking the derivative of the outer function and multiplying it by the derivative of the inner function.
Let's break it down step by step:
1. Start by applying the chain rule. Take the derivative of the outer function (cos^5(u)), where u represents the expression inside the cosine function, and multiply it by the derivative of the inner function (4x-19).
2. The derivative of cos^5(u) can be found using the power rule for derivatives and the chain rule. The power rule states that if you have a function raised to a power, you bring down the exponent as a coefficient and reduce the power by 1. Therefore, the derivative of cos^5(u) is 5cos^(5-1)(u) * (-sin(u)).
3. Now, substitute the inner function back in. We have 5cos^4(4x-19) * (-sin(4x-19)).
So, your answer is indeed f'(x) = -5cos^4(4x-19) * sin(4x-19). However, it seems there might be a discrepancy in your answer regarding the coefficient, as it should be -5 instead of -20.