Question

why can't you just take the power rule for trig functions to find the derivative

like for sin^2 the derivative is not 2sin but rather u take the product rule and get 2sincos

Answer 1

its not the product rule; its the chainrule. It is because the trig functions are ... well, "functions" and not simply variables

Answer 2

sin^2 = (sin)^2

[(sin)^2] = 2(sin)* [sin]'
[(sin)^2] = 2sin cos

Answer 3

[f(g(x))]' = f'(g(x)) * g'(x) * x'