Question

How do you find the derivative of `sin^2(2x)`?

Answer 1

I got: `f'(x)=4sin(2x)cos(2x)`

Explanation:

Here I would try to use the Chain Rule applied to three functions one nested into the other:
the first and all-embracing function is `()^2`; the next one is the `sin` function and the last one the `2x` function.

I will use the Chain Rule deriving each one as if alone (regardless of the argument) and I will multiply each individual derivative together using, as visual help, a sequence of red-blue-green colors to identify each derivative:

`f'(x)=color(red)(2sin^(2-1)(2x))*color(blue)(cos(2x))*color(green)(2)`
giving:
`f'(x)=4sin(2x)cos(2x)`

This function can be compressed (giving: `2sin(4x)`) using a trigonometric identity but I do not want to confuse the procedure.