how does thederivative of -2xcos(x^2)= -4xcos(x^2) -2sin(x^2) and not -4x^2cos(x^2) -2sin(x^2)?

Answer 1

Huh? If you let
u = x^2, du = 2x dx, so
∫ -2x cos(x^2) dx = ∫ -cosu du = -sinu = -sin(x^2)

I suspect a typo.

Answer 2

if y = (-2x)(cos(x^2))
then the derivative dy/dx = (-2x)(-2x)(sin(x^2)) + (-2)cos(x^2) , by the product rule
= (4x^2)sin(x^2) - 2cos(x^2)

Neither of your choices matches that, so like oobleck, I suspect a typo.

Answer 3

oops. I thought it said anti-derivative. Mental typo!