Proof of Derivative of Cosec x

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

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

Derivative of Cosec x by First Principle of Derivative

To prove derivative of cosec x using First Principle of Derivative, we will use basic limits and trigonometric formulas which are listed below:

• cosec x = 1/sin x
• lim_h→0 (sin(x + h) – sin x)/h = cos x
• cot x = cos x/sin x

Let’s start the proof for the derivative of cosec x

By First Principle of Derivative

Let y = cosec x

y = 1/sin x

⇒ y’ = d/dx (1/sin x)

⇒ y’ = lim _h→0 (1/sin(x + h) – 1/sin x) / ((x + h) – x)

⇒ y’ = lim _h→0 ((sin x – sin(x + h)) / (sin x × sin(x + h))) / h

⇒ y’ = lim _h→0 (sin x – sin(x + h)) / (h × sin x × sin(x + h))

⇒ y’ = lim _h→0 – (sin(x + h) – sin x) / (h × sin x × sin(x + h))

⇒ y’ = lim _h→0 – (sin(x + h) – sin x) /h × lim _h→0 1 /(sin x × sin(x + h))

⇒ y’ = -cos x × 1 / sin² x

⇒ y’ = -cos x / sin x × 1 / sin x

⇒ y’ = -cot x × cosec x

Therefore, the differentiation of cosec x is – cosec x cot x.

Derivative of Cosec x by Quotient Rule

To prove the derivative of cosec x using the Quotient rule, we will use basic derivatives and trigonometric formulas which are listed below:

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

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

y = cosec x

⇒ y = 1/sin x

⇒ y’ = d/dx (1/sin x)

Applying quotient rule

y’ = ((d/dx) (1) × sin x – 1 × (d/dx)(sin x))/sin² x

⇒ y’ = ((0) × sin x – (1) × (cos x))/sin² x

⇒ y’ = -cos x/(sin x)²

⇒ y’ = -cot x × cosec x

Therefore, the differentiation of cosec x is – cosec x cot x.

Derivative of Cosec x by Chain Rule

To prove derivative of cosec x we will use chain rule and some basic trigonometric identities and limits formula. The trigonometric identities and limits formula which are used in the proof are given below:

• cot x = cos x / sin x
• cosec x = 1 / sin x
• (d/dx) sin x = cos x

Let’s start the proof for the differentiation of the trigonometric function cosec x

(d/dx) cosec x = (d/dx) (1 / sin x)

Using chain rule

(d/dx) cosec x = (-1 / sin²x) (d/dx) sin x

⇒ (d/dx) cosec x = (-1 / sin²x) cos x

⇒ (d/dx) cosec x = -(1 / sin x) (cos x / sin x)

⇒ (d/dx) cosec x = – cosec x cot x

Therefore, the differentiation of cosec x is – cosec x cot x.

Also, Check

• Derivative of Inverse Trigonometric Functions
• Differentiation Formulas
• Differentiation of Trig Functions

Derivative of Cosec x is -Cot x Cosec x. The derivative of cosec x is represented by the d/dy(cosec x). It explains about the slope of the graph of cosec x. Cosecant Functions are denoted as csc or cosec and defined as the reciprocal of the sine function i.e., 1/sin x.

In this article, we will discuss all the topics related to the derivative of cosec x including its proof using various methods. Let’s start our learning on the topic of Derivative of Cosec x.