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

Dec 18, 2014

The derivative of ${\cos}^{3} \left(x\right)$ is equal to:
$- 3 {\cos}^{2} \left(x\right) \cdot \sin \left(x\right)$
You can get this result using the Chain Rule which is a formula for computing the derivative of the composition of two or more functions in the form: $f \left(g \left(x\right)\right)$.
You can see that the function $g \left(x\right)$ is nested inside the $f \left(\right)$ function.
Deriving you get:
derivative of $f \left(g \left(x\right)\right)$ --> $f ' \left(g \left(x\right)\right) \cdot g ' \left(x\right)$

In this case the $f \left(\right)$ function is the cube or ${\left(\right)}^{3}$ while the second function "nested" into the cube is $\cos \left(x\right)$.

First you deal with the cube deriving it but letting the argument $g \left(x\right)$ (i.e. the $\cos$) untouched and then you multiply by the derivative of the nested function.

Which is equal to: $- 3 {\cos}^{2} \left(x\right) \cdot \sin \left(x\right)$