You can use the product rule straight-away:
I need help finding the derivative of:
sin^2(x)
I believe I have to use the half angle identity...
if this is true, then would it be:
1/2 (1-cos(2x)) ?
would I need to take the derivative of this after the identity?
If this holds true, would be 2sin(2x) ???
1/2(1- cos(2x))= 1/2- 1/2 cos(2x). Of course, the derivative of 1/2 is 0. Using the chain rule, the derivative of cos(2x) is - sin(2x)(2x)'= -2 sin(2x). Of course, then, the derivative of -1/2 cos(2x) is (-1/2)(-2) sin(2x)= sin(2x). The derivative of 1/2(1- cos(2x)) is sin(2x), not 2 sin(2x).
But I would have done this using the chain rule directly- Let u= sin(x) and y= u^2= sin^2(x). Then .
Or, more simply, since the derivative of is 2x, the derivative of is 2x, the derivative of is 2 sin(x) times the derivative of sin(x), which is cos(x).
Of course, those are the same answer- sin(2x)= 2 sin(x)cos(x).