Examples Using Derivative of Cosec x

Some examples on Using Derivative of Cosec x are,

Example 1: Find the derivative of cosec 4x.

Solution:

Let y = cosec 4x

y’ = (d/dx) [cosec 4x]

Applying chain rule

y’ = (d/dx) [cosec 4x].(d/dx) (4x)

⇒ y’ = (-cot 4x × cosec 4x) × 4

⇒ y’ = -4 × cot 4x × cosec 4x

Example 2: Evaluate the derivative f(x) = (x³ + 5x² + 2x + 7) × cosec x.

Solution:

f(x) = (x³ + 5x² + 2x + 7) × cosec x

⇒ f'(x) = (d /dx)[(x³ + 5x² + 2x + 7) × cosec x]

Applying product rule

⇒ f'(x) = (d /dx)[(x³ + 5x² + 2x + 7)] × cosec x + (x³ + 5x² + 2x + 7) × (d /dx)[cosec x]

⇒ f'(x) = (3x² + 10x + 2) × cosec x + (x³ + 5x² + 2x + 7) × (-cot x × cosec x)

Example 3: Determine the second derivative of cosec x.

Solution:

The first derivative of cosec x is -cosec x cot x.

To determine the second derivative of cosec x, we differentiate -cosec x cot x using the product rule.

(cosec x)” = (-cosec x cot x)’

⇒ (-cosec x)’ cot x + (-cosec x) (cot x)’

⇒ cosec x cot x cot x + (-cosec x) (-cosec²x)

⇒ cosec x (cot²x + cosec²x)

Second derivative of cosec x is cosec x (cot²x + cosec²x).

Example 4: Find the derivative of cosec^-1 x.

Solution:

d/dx[cosec^-1 x] = -1 / (|x| × sqrt(x² – 1)), from formula

Example 5: Evaluate the derivative cosec 5x + x × cosec x.

Solution:

Let z = cosec 5x + x × cosec x

Differentiating

z’ = (d/dx) [cosec 5x + x × cosec x]

⇒ z’ = (d/dx) cosec 5x + (d/dx)[x × cosec x]

Applying chain rule and product rule

z’ = -5 × cot 5x × cosec 5x + (d/dx)(x) × cosec x + x × (d/dx)(cosec x)

⇒ z’ = -5 × cot 5x × cosec 5x + cosec x + x × (-cot x × cosec x)

⇒ z’ = -5 × cot 5x × cosec 5x + cosec x – x × cot x × cosec x

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.