What Is The Differentiation Of Cot X

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In calculus, differentiation is a fundamental concept used to find the rate of change of a function. One of the most commonly asked differentiation problems involves trigonometric functions, including cotangent (cot x).

This topic will explain:
✔ The definition of cot x
✔ The formula for differentiating cot x
✔ A step-by-step derivation of d/dx (cot x)
✔ Examples of differentiation involving cot x
✔ The real-world applications of cotangent differentiation

Let’s explore the differentiation of cot x in detail!

1. What Is Cot x?

Cotangent, written as cot x, is a trigonometric function defined as:

cot x = frac{cos x}{sin x}

This function is the reciprocal of the tangent function:

cot x = frac{1}{tan x}

Understanding this definition helps in finding the derivative of cot x more easily.

2. What Is the Differentiation Formula of Cot x?

The derivative of cot x with respect to x is given by the formula:

frac{d}{dx} (cot x) = -csc2 x

where csc x (cosecant) is another trigonometric function defined as:

csc x = frac{1}{sin x}

So, the derivative of cot x results in a negative squared cosecant function.

3. Step-by-Step Derivation of d/dx (cot x)

To find the derivative of cot x, we use the quotient rule, since cot x is written as cos x / sin x.

Step 1: Use the Quotient Rule

The quotient rule states that if a function is in the form f(x) / g(x), then its derivative is:

frac{d}{dx} left( frac{f(x)}{g(x)} right) = frac{f'(x)g(x) – f(x)g'(x)}{g2(x)}

For cot x = cos x / sin x, let:
✔ f(x) = cos x (numerator)
✔ g(x) = sin x (denominator)

Now, differentiate both:
✔ f'(x) = -sin x
✔ g'(x) = cos x

Step 2: Apply the Quotient Rule

frac{d}{dx} (cot x) = frac{(-sin x) cdot sin x – (cos x) cdot (cos x)}{(sin x)2}
= frac{-sin2 x – cos2 x}{sin2 x}

Step 3: Use the Pythagorean Identity

We use the identity:

sin2 x + cos2 x = 1

So,

-sin2 x – cos2 x = -1

Thus,

frac{d}{dx} (cot x) = frac{-1}{sin2 x}

Since 1 / sin² x = csc² x, we get:

frac{d}{dx} (cot x) = -csc2 x

This confirms the differentiation formula! ✅

4. Examples of Differentiating Cot x

Example 1: Basic Differentiation

Find d/dx (cot x)

Solution:

frac{d}{dx} (cot x) = -csc2 x

That’s the direct formula!

Example 2: Differentiating a Function Involving Cotangent

Find d/dx (cot 3x)

Solution:

Use the chain rule, where cot u has derivative -csc² u · du/dx.

Here, u = 3x, so du/dx = 3.

frac{d}{dx} (cot 3x) = -csc2 (3x) cdot 3
= -3 csc2 (3x)

Example 3: Finding the Tangent Line

Find the equation of the tangent line to y = cot x at x = frac{pi}{4}.

Solution:

Step 1: Differentiate y = cot x

frac{dy}{dx} = -csc2 x

Step 2: Find the slope at x = π/4

frac{dy}{dx} Bigg|_{x = frac{pi}{4}} = -csc2 left(frac{pi}{4}right)

Since sin(π/4) = 1/√2, we get:

csc (π/4) = frac{1}{sin (π/4)} = sqrt{2}
csc2 (π/4) = 2

So,

frac{dy}{dx} = -2

Step 3: Use the point-slope equation y – y₁ = m(x – x₁)

At x = π/4, cot (π/4) = 1, so y₁ = 1.

y – 1 = -2 left(x – frac{pi}{4}right)
y = -2x + frac{pi}{2} + 1

This is the equation of the tangent line.

5. Real-World Applications of Differentiating Cot x

Differentiating cotangent has various applications in mathematics, physics, and engineering.

✔ Physics – Used in wave motion and oscillation problems.
✔ Engineering – Helps in analyzing slopes in mechanical structures.
✔ Economics – Used in modeling rate changes in economic trends.

For example, in optics, cotangent functions are used to describe light wave behavior in lenses and mirrors.

The differentiation of cot x follows a standard formula:

frac{d}{dx} (cot x) = -csc2 x

✔ Definition – Cotangent is cos x / sin x.
✔ Formula – The derivative of cot x is -csc² x.
✔ Step-by-Step Derivation – Using the quotient rule, we get the result.
✔ Examples – Differentiating cot x, cot 3x, and finding tangent lines.
✔ Applications – Used in physics, engineering, and economics.

Understanding the derivative of cot x is essential for solving advanced calculus problems and real-world applications. 🎯