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

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justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

The derivative of f(x) = (cos^ 2 x)* ln (x^2) has to be found.

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

=> f(x) = (cos x)^2 * ln (x^2)

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

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

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

The derivative of the given function is (cos x)^2 * [2/x] - sin (2x)* ln (x^2)

giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

Since the given function is the result of composition of functions, we'll determine the 1st derivative using 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 first derivative of the 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

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