A short note on derivative of Cot function

Cot is the short for the trigonometric function cotangent. It is the complementary function of the tan also called the tangent function. Cot is defined as the ratio of the adjacent side to the opposite side of the acute angle in a right triangle. It is also the reciprocal of the tan function.

Therefore symbolically it can be written like this:
Cot(x) = adjacent side/opposite = 1/tan(x)
We can graph the cot function using a table of values as follows:
X -pi -3pi/4 -pi/2 -pi/4 0 Pi/4 Pi/2 3pi/4 pi
Cot(x) Inf 1 0 -1 Inf 1 0 -1 inf

The graph would look as follows:

 From the graph we note that the cot function is not defined at the points –pi, 0 and pi. That is because at these points, the tan function has the values of 0,0 and 0 respectively. Since the cot function is the reciprocal of tan function, the cot is not defined at these points.

Derivative Cot X  function:

The above graph is of the function y = Cot(x). Now the derivative of Cot X would be the slope of tangent to the above curve at any point x. Therefore for example if we were to find the derivative of Cotx at the point x = pi/4, then the graph of the tangent line to the curve at that point would like this:

The blue line in the above picture is the tangent to the cot function at the point x= pi/4. The slope of this tangent is the derivative of cot x. In other words we can also say that the derivative of the cot function is the rate of change of cot function at a given point. This is also called the instantaneous rate of change of cot function at a point x=a. (In this case it is x = pi/4). Another way of stating the same thing is like this: the gradient of the curve of the function y = cot x at a point x=a, is called the derivative of the function at that point.

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The derivative of the function y = cot x at the point x = 0 can be found by making a tangent to the curve at the point x=0. Again from the above graph we see that, at x=0, the tangent would be vertical. So slope of tangent = not defined. Therefore derivative at this point is not defined.