Question

How do you differentiate `f(x) = sin(2x)cos(2x)` using the product rule?

Answer 1

See the explanation section below.

Explanation:

Differentiate `f(x) = sin(2x)cos(2x)`

using the product rule
I use the order: the derivative of a product of functions is the derivative of the first times the second, plus: the first times the derivative of the second.
`d/dx(FS) = F'S+FS'`

Note that we shall need the chain rule for the derivatives of `sin(2x)` and `cos(2x)`

You may choose to write `F`, `S`, `F'` and `S'` before using the formula. I do not have that habit, so

`f'(x) = [cos(2x)(2)]cos(2x)+sin(2x)[-sin(2x)(2)]`

`= 2cos^2(2x)-2sin^2(2x)`

Your teacher/textbook may well prefer to rewrite this answer using `cos(2theta) = cos^2theta - sin^2theta`, to get

`= 2[cos^2(2x)-sin^2(2x)]`

`= 2cos(4x)`

Although if you're going to do that, I suggest

Rewriting the function

Use `sin(2theta) = 2sintheta costheta` to rewrite `f(x)` as

`f(x) = sin(2x)cos(2x) = 1/2sin(4x)`.

Now we do not need the product rule, only the chain rule (which we needed in the other method also).