Differentiate f(x)=cos^3(x^3).

We'll use the chain rule to differentiate the given
function:

f'(x) =
{[cos(x^3)]^3}'

We'll differentiate applying the power rule
first, then we'll differentiate the cosine function and, in the end, we'll differentiate
the variable x^3.

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

f'(x) =
-9x^2[cos(x^3)]^2*[sin(x^3)]

We can re-write
[cos(x^3)]^2 = 1 - [sin(x^3)]^2

f'(x) = -9x^2{1 -
[sin(x^3)]^2}*[sin(x^3)]

f'(x) =
9x^2*[sin(x^3)]^3 - 9x^2*[sin(x^3)]