quick differentiation question about sin^2(x)

Ignorant question here, but I was looking at this example i have, needing to take the derivative:

f'(x) = sin^2(x) + cos(x)

= 2sin(x)*d/dx(sin x) - sin(x)

= 2sin(x)*cos(x) - sin(x)

and so on... I'm wanting to know why the 2sin(x) isn't where the differentiation stops for sin^2(x)? there's the 2sin(x), and then the d/dx(sinx) added in as well. The cos(x) on the right side doesn't have any extra steps for it. Hope this makes sense, and i realize this is a silly question but I just wanted the clarification.