Solved Examples on Derivative of Cot x

Some examples related to Derivative of Cot x are,

Example 1: Find the derivative of cot²x.

Solution:

Let f(x) = cot²x = (cot x)²

By using power rule and chain rule,

f'(x) = 2 cot x · d/dx(cot x)

We know that the derivative of cot x is -cosec²x. So

f'(x) = -2 cot x ·cosec²x

Example 2: Differentiate tan x with respect to cot x.

Solution:

Let v = tan x and u = cot x. Then dv/dx = sec²x and du/dx = -cosec²x.

We have to find dv/du. We can write this as

dv/du = (dv/dx) / (du/dx)

dv/du = (sec²x) / (-cosec²x)

dv/du = (1/cos²x) / (-1/sin²x)

dv/du = (-sin²x) / (cos²x)

dv/du = -tan²x

Example 3: Find the derivative of cot x · csc2x

Solution:

Let f(x) = cot x · cosec²x

By product rule,

f'(x) = cot x·d/dx (cosec²x) + cosec²x·d/dx(cot x)

f'(x) = cot x·(2 cosec x) d/dx (cosec x) + cosec²x (-cosec²x) (by chain rule)

f'(x) = 2 cosec x cot x (-cosec x cot x) – cosec⁴x

f'(x) = -2 cosec²x cot²x – cosec⁴x

Derivative of Cot x is -cosec²x. It refers to the process of finding the change in the sine function with respect to the independent variable. Derivative of cot x is also known as differentiation of cot x which is the process of finding rate of change in the cot trigonometric function.

In this article, we will learn about the derivative of cot x and its formula including the proof of the formula using the first principle of derivatives, quotient rule, and chain rule as well.