Proof of Derivative of Cot x

The derivative of cot x can be proved using the following ways:

• By using First Principle of Derivative
• By using Quotient Rule
• By using Chain Rule

Derivative of Cot x by First Principle of Derivative

Let’s start the proof for derivative of Cot x:

Let f(x) = Cot x

By the First Principle of Derivative

f'(x)= lim h→0 f(x+h)-f(x)/h

= lim h→0 cot(x+ h)- cot x/ h

= lim h→0 [cos(x+h)/sin(x+h)- cos x/ sin x]/h

= lim h→0 sin x cos(x+h)-cos x sin (x+h) / sin(x+h) sin x. h

=lim h→0 sin [x-(x+h) / sin(x+h).sin x .h

= lim h→0 – sin h/h lim h→0 1/sin (x+h)sin x

= -1 × 1/sinx. sinx

= -1/ sin²x

= -cosec²x

Derivative of Cot x by Quotient Rule

To find the derivative of cot x using the quotient rule of derivative we have to use the following mentioned formulas

• (d/dx) [u/v] = [u’v – uv’]/v²
• sin²(x)+ cos²(x)= 1
• cot x = cos x / sin x
• cosec x = 1 / sin x

Let’s start the proof of the derivative of cot x

f(x) = cot x = cos(x)/sin(x)

u(x) = cos(x) and v(x)=sin(x)

u'(x) = -sin(x) and v'(x)=cos(x)

v²(x) = sin²(x)

f'(x) = {-sin(x).sin(x) – cos(x).cos(x)}/sin²(x)

f'(x) = -sin²(x)-cos²(x)/sin²(x)

f'(x) = -sin²(x)+cos²(x)/sin²(x)

By one of the trigonometric identities, cos²x + sin²x = 1.

f'(x) = – 1/ sin²(x)

d/dx cot(x) = -1 /sin²(x) = -cosec²(x)

Therefore, differentiation of cot x is -cosec²x.

Derivative of Cot x by Chain Rule

Assume y = cot x then we can write y = 1 / (tan x) = (tan x)^-1. Since we have power here, we can apply the power rule here. By power rule and chain rule,

y’ = (-1) (tan x)^-2·d/dx (tan x)

The derivative of tan x is, d/dx (tan x) = sec²x

y= cot x

y’ = -1/tan²x·(sec²x)

y’ = – cot²x·sec²x

Now, cot x = (cos x)/(sin x) and sec x = 1/(cos x). So

y’ = -(cos²x)/(sin²x) · (1/cos²x)

y’ = -1/sin²x

Since, reciprocal of sin is cosec. i.e., 1/sin x = cosec x. So

y’ = -cosec²x

Hence proved.

Also Read,

• Differentiation of Trigonometric Function
• Differentiation Formulas
• Derivative of Root x

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.