What is the derivative of $cos^2(x^3)$ ?

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Answer 1 · 2016-01-22T13:00:46

$f'(x) = -6 x^2 cos (x^3) sin (x^3)$

$f(x) = cos^2(x^3)$

Let's break your function down as a chain of functions:

$f(x) = [color(blue)(cos (x^3))]^2 = color(blue)(u)^2$

where

$u = cos(color(violet)(x^3)) = cos(color(violet)(v))$

where

$v = x^3$

Thus, the derivative of $f(x)$ is:

$f'(x) = [ u^2 ]' * u' = [u^2]' * [cos v]' * v'$

Now, let's compute those three derivatives:

$[u^2]' = 2u = 2 cos x^3$

$[cos v]' = - sin v = - sin x^3$

$[ v]' = [x^3]' = 3x^2$

Thus, you can compute your derivative as follows:

$f'(x) = [u^2]' * [cos v]' * v'$

$= 2 cos x^3 * (- sin x^3) * 3x^2$

$= -6 x^2 cos (x^3) sin (x^3)$

What is the derivative of cos^2(x^3)cos^2(x^3)?