Differentiate g(x) = x^3 * cos^2 x

To calculate the first derivative of the given function, we'll
use the product rule and the chain rule:

g'(x) = x^3 * (cos
x)^2

We'll have 2 functions f and
h:

(f*h)' = f'*h + f*h'

We'll put f =
x^3 => f' = 3x^2

We'll put h = (cos x)^2 => h' = 2(cos
x)*(cos x)'

h' = -2(sin x)*(cos x)

h' =
- sin 2x

We'll substitute f,h,f',h' in the expression of
(f*h)':

(f*h)' = 3x^2*(cos x)^2 - x^3*(sin
2x)

We'll factorize by
x^2:

g'(x) = x^2[3*(cos x)^2 - x*(sin
2x)]