How to differentiate cot x

d/dx (cot x) = d/dx (cos x / sin x)
= sin x * - sin x - cos x * cosx / sin^2x
= - (sin^2x + cos^2x) / sin^2x
= - 1 / sin^2x
= - csc^2x

use the product rule on cos(x) / sin(x)
you'll get - csc^2(x)