Derivatives of Cot functions for some special values

We can find the derivative of the cot function using the quotient rule of differentiation. The quotient rule can be states as follows: For a function f such that f(x) = u/v, where u and v are both function of x, the derivative of f(x) is given by the formula:
f’(x) = (v*u’ – u*v’)/v^2, where u’ is derivative of function u and v’ is derivative of function v.
We know that, cot x = cos x / sin x,
Therefore, (d/dx) cot x = (d/dx) (cos x / sin x),
= [sin x (d/dx) (cos x) – cos x (d/dx) (sin x)]/sin^2 (x)  … using the quotient rule stated above.
=[ sin x (-sin x) – cos x * cos x]/sin^2(x)
= [-sin^2(x) – cos^2(x)]/sin^2(x)
= -[sin^2(x) + cos^2(x)]/sin^2(x)
= -1/sin^2(x)
= -csc^2(x)
Therefore the derivative of cot x is –csc^2(x)

The above derivative can also be proved using the definition of derivative which is like this:
F’(x) = lim(t->x) [(f(t) – f(x)]/(t-x), for the function y = f(x) = cot (x) the definition would be like this:
(d/dx) cot x = lim(t->x) [cot t – cot x]/(t-x)
= lim(t->x) [1/tan t – 1/tan x]/(t-x)
  = lim(t->x) [tan x – tan t]/[(tan t * tan x)(t-x)]
= lim(t->x) {[tan x – tan t]/(t-x)} * 1/[tan t * tan x]
= lim(t->x) {-[tan t – tan x]/(t-x)} * lim(t->x) 1/[tan t * tan x]
= lim(t->x) {-[ tan(t-x) (1+tan t tan x)]/(t-x)} * 1/tan^2(x)
We know that lim(t->x) of tan(t-x)/(t-x) = 1. Therefore,
= -{1+tan^2(x)} *1/tan^2(x)
= - sec^2(x)/tan^2(x)
= [-1/cos^2(x)]/[sin^2(x)/cos^2(x)] … here the cos^2(x) cancel each other out.
= -1/sin^2(x)
= -csc^2(x)

Derivative of cot-x:
For finding the derivative of cot(-x) we use the chain rule.
Let –x = u, then du/dx = -1.
Cot (-x) = cot u
(d/du) cot u = - csc^2(u) …  from what we proved above.
Using the chain rule we have:
(d/dx) cot (-x) = (d/dx) cot u * du/dx = -csc^2(u) * (-1)
= -csc^2(-x) * (-1)
= csc^2(-x)
Thus we see that the derivative of cot(-x) is csc^2(-x)

Derivative of cot^-x (k):
Cot^(-x) (k) where k is some constant can be written like this also
= (cot(k))^(-x)
= 1/(cot k)^x
= (1/cot k)^x
Since k is a constant, cot k is also a constant and in turn, 1/cot k is also constant. So let 1/cot k = some constant say p. Therefore,
= p^x is the function.
(d/dx) p^x = p^x ln p
= (1/cot k)^x ln(1/cot k)
That is the derivative of the above function.