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# What is the Derivative of Cot x: Easy Guide

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Calculus often appears complex, but breaking down derivatives is a vital step in mastering it. Think of cot x as a ratio in a right triangle, the side next to x divided by the side opposite x, or cos of x divided by sin of x. In this blog, we’re going to find the derivative of cot x.  By the time you reach the end, you’ll have a clear grasp of this essential concept in mathematics.

### Finding the Derivative

When we want to find the derivative of cot x, we’re basically figuring out how it changes as x changes. The result is -1 times the square of cosec x, which might sound complex, but it’s just a way of measuring how the function varies with x.

Recommended Reading: What is the Derivative of sec x?

#### The formula for differentiation of cot x is,

d/dx (cot x) = -cosec2x (or)

(cot x)’ = -cosec2x

Let’s prove it step-by-step:

##### Step 1: Express cot x in terms of sine and cos:

cot x = cos x / sin x.

##### Step 2: Identify u and v for the quotient rule:

Let u = cos x.

Let v = sin x.

##### Step 3: Find the derivatives of u and v

a. Find du/dx:

Derivative of cos x is -sin x.

b. Find dv/dx:

Derivative of sin x is cos x.

##### Step 4: Apply the quotient rule

We know that cot x = (cos x)/(sin x).

So we assume that cot x = y

y = (cos x)/(sin x).

##### Step 5: Then by quotient rule derivative of y = y’,

y’ = [ sin x · d/dx (cos x) – cos x · d/dx (sin x)] / (sin^2 x)

= [sin x · (- sin x) – cos x (cos x)] / (sin^2 x)

= [-sin^2 x – cos^2 x] / (sin^2 x)

= -[-sin^2 x + cos^2 x] / (sin^2 x)

By one of the Pythagorean identities, cos^2 x + sin^2 x = 1. So

y’ = -1 / (sin^2 x) = -cosec^2 x

Recommended Reading: Understanding the Derivative of 2/x

### Conclusion

Tackling calculus can feel like navigating through a maze, but understanding derivatives is like finding the key to unlock it. We’ve taken a deep dive into the derivative of cot x, picturing it as a ratio in a right triangle—cos of x divided by sin of x—which sheds light on its behavior as x changes. By following along step by step, you’ve unlocked a fundamental puzzle piece in the world of math.

#### Moonpreneur

Moonpreneur is an ed-tech company that imparts tech entrepreneurship to children aged 6 to 15. Its flagship offering, the Innovator Program, offers students a holistic learning experience that blends Technical Skills, Power Skills, and Entrepreneurial Skills with streams such as Robotics, Game Development, App Development, Advanced Math, Scratch Coding, and Book Writing & Publishing.
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