Find f'(x) if f(x)=sin^3(4x)
A. 4cos^3(4x)
B. 3sin^2(4x)cos(4x)
C. cos^3(4x)
D. 12sin^2(4x)cos(4x)
E. None of these
I got D using the chain rule?
f = u^3, so
f' = 3u^2 u' = 3sin^2(4x) cos(4x) (4)
= 12 sin^2(4x) cos(4x)
To find the derivative f'(x) of the function f(x) = sin^3(4x), you can use the chain rule.
The chain rule states that for a composition of functions, the derivative of the outer function multiplied by the derivative of the inner function. In this case, the outer function is the power of 3, and the inner function is sin(4x).
First, let's find the derivative of the outer function: d/dx(sin^3(θ)) = 3sin^2(θ)cos(θ). Note that we can substitute θ with 4x because θ = 4x in this case.
Now, let's apply the chain rule. f'(x) = derivative of the outer function * derivative of the inner function.
f'(x) = 3sin^2(4x)cos(4x) * 4 = 12sin^2(4x)cos(4x).
Therefore, the correct answer is D.