Differentiate y=cos²x

Use the chain rule.

cos^2(x) is the same thing as (cos x)^2.

f'(g(c))*g'(c), so x^2 is f and cos x is g

derivative of f = f' = 2x, so f'(g(c)) = 2(cos x)
derivative of g = g' = -sin x, so g'(c) = -sin x

therefore, the derivative of y=cos^2(x) = -2(cos x)(sin x)

y: (cos x)²
dy/dx: 2(cos x)1. - sin x [That's 2(cos x) to the power of one]
dy/dx: -2sin x.cos x

y'[x] = -2 Cos[x] Sin[x]

Use the chain rule

y' = 2cosx(-sinx) = -sin(2x)