What is the first derivative of cos^3(x^3+13) ?

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

f'(x) =
{[cos(x^3+13)]^3}'

We'll calculate the first
derivative applying the power rule first, then we'll differentiate the cosine function
and, in the end, we'll differentiate the expression
x^3+13.

f'(x) =
3[cos(x^3+13)]^2*[-sin(x^3+13)]*(x^3+13)'

f'(x) =
3[cos(x^3+13)]^2*[-sin(x^3+13)]*(3x^2)

f'(x)
= -9x^2*[cos(x^3+13)]^2*[sin(x^3+13)]

We can
re-write [cos(x^3+13)]^2 = 1 -
[sin(x^3+13)]^2

f'(x) =
-9x^2*[sin(x^3+13)]*{1 - [sin(x^3+13)]^2}