Question

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

Answer 1

The derivative of `cos^3(x)` is equal to:
`-3cos^2(x)*sin(x)`
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(g(x))`.
You can see that the function `g(x)` is nested inside the `f( )` function.
Deriving you get:
derivative of `f(g(x))` --> `f'(g(x))*g'(x)`

In this case the `f( )` function is the cube or `( )^3` while the second function "nested" into the cube is `cos(x)`.

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

Which is equal to: `-3cos^2(x)*sin(x)`