Question

simplify:
sin 2x cos 2x
using derivatives.

Answer 1

That makes no sense. Can you please reword that?

Answer 2

do you want us to help you take the derivative of that function?

Answer 3

exactly

Answer 4

i would clean it up a little first using the double angle formula:
\[\sin(2x)\cos(2x) = \frac{1}{2}(2\sin(2x)\cos(2x)) = \frac{1}{2}\sin(4x)\]

Answer 5

Now taking the derivative is just an application of the chain rule, instead of chain and product rule.

\[\frac{d}{dx}\left(\frac{1}{2}\sin(4x)\right) = \frac{1}{2}\cos(4x)*4 = 2\cos(4x)\]

Answer 6

can i ask how come a one half existed in the simplification?

Answer 7

i multiplied by a clever form of "one":
\[\sin(2x)\cos(2x) = 1*\sin(2x)\cos(2x) = \frac{2}{2}\sin(2x)\cos(2x) = \]\[\frac{1}{2}(2\sin(2x)\cos(2x))\]