Differentiate f(x) =( cosx)^3 * (lnx)^2

Given the function f(x) = (cosx)^3 *
(lnx)^2.

We need to find the first derivative of
f(x).

We notice that f(x) is a product of two
functions.

Then we will use the product rule to find the first
derivative.

Let f(x) = u * v    such
that:

u= (cosx)^3     ==>   u' = -3(cosx)^2 *
sinx

v = (lnx)^2      ==>  v' = 2lnx *
1/x.

Then we will use the product
rule:

==> f'(x) = u'*v +
u*v'

Let us substitute with u, u' , v', and
v.

           = (-3sinx(cosx)^2 *(lnx)^2 + (cosx)^3
(2lnx)/x

            = (cosx)^2 (lnx) [ -3sin*lnx +
2cosx)/x)

==> f'(x) = (cosx)^2 (lnx) [
-3sin*lnx + 2cosx)/x)