The derivative of cot(x) is -csc^2(x). The derivatives of the secant, cosecant and cotangent functions are based on the derivatives of their reciprocal trigonometric functions. For example, the derivative of cotangent is equal to the derivative of one over the tangent of x.
The derivative of one over the tangent of x can be simplified as the difference of the derivative of the tangent of x subtracted from the tangent of x times the derivative of one, with the whole difference divided by tangent squared of x. From there, the formula becomes the negative secant squared of x, divided by the tangent squared of x.