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

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We have to find the first derivative of [cos ( x^3 + 13)]^3.

Here we use the chain rule to arrive at the solution.

([cos ( x^3 + 13)]^3 )'

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

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

**The required derivative of [cos ( x^3 + 13)]^3 is -9*x^2* [cos ( x^3 + 13)]^2*sin ( 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}**