Differentiate f(x) = (cos^ 2 x)* ln (x^2).

Since we have to determine the first derivative of a composed
function, we'll apply the chain rule and also the product
rule.

f'(x) = [(cos x)^2]'* ln (x^2) + (cos x)^2* [ln
(x^2)]'

We'll apply the chain rule for the
terms:

[(cos x)^2]' = 2 cos x*(cos
x)'

[(cos x)^2]' = - 2cos x*sin x

 [ln
(x^2)]' = (x^2)'/x^2

 [ln (x^2)]' =
2x/x^2

We'll simplify and we'll
get:

 [ln (x^2)]' = 2/x

The result of
sifferentiating the given function is:

f'(x) =  - 2cos x*sin x*ln
(x^2) + 2(cos x)^2/x

f'(x) = -sin 2x*ln (x^2) + 2(cos
x)^2/x