Final answer:
The derivative of the function g(x) = 9 cos³(x) is -27cos²(x)sin(x), which is found using the chain rule and the power rule of differentiation.
Step-by-step explanation:
To find the derivative of the function g(x) = 9 cos³(x), we need to use the chain rule and the power rule of differentiation.
Let's denote u = cos(x). Thus, g(x) = 9u³. First, we'll differentiate u³ with respect to u to get 3u². Then we need to multiply by the derivative of u with respect to x, which is -sin(x). Therefore, the derivative of g with respect to x is:
g'(x) = 9 * 3u² * (-sin(x))
= -27u²sin(x)
= -27cos²(x)sin(x)
The final answer is the derivative of g(x), which is -27cos²(x)sin(x).