# Derivative of coth x

Learn how to calculate the derivative of a coth x by first principle with easy steps. Also verify the derivative of coth x by using chain rule and product rule.

Alan Walker-

Published on 2023-05-26

## What is the Derivative of coth x?

Derivatives have a wide range of applications in almost every field of engineering and science. The coth x derivative can be calculated by following the differentiation rules. Or, we can directly find the differentiation of coth x by applying the first principle of differentiation. In this article, you will learn what the derivative of coth x is and how to calculate the derivative coth x by using different approaches.

## What is the derivative of coth x?

The derivative of coth(x) with respect to the variable 'x' is expressed as d/dx(coth x) and is equal to the negative square of cosech^2x. This formula represents the rate of change of the hyperbolic function coth x, which is the ratio of the hyperbolic cosine (cosh x) and hyperbolic sine (sinh x) functions. In trigonometry, this ratio corresponds to the ratio of the adjacent and opposite sides of a right triangle. Mathematically, it can be written as:

$$\coth x =\frac{\cosh x}{\sinh x}$$

## Derivative of coth x formula

The formula for finding the differentiation of coth x is equal to the negative of the csch squared function, which is denoted by csch^2x. Mathematically, this can be written as:

$$\frac{d}{dx}(\coth x)= -\DeclareMathOperator{\csch}{csch}\csch^2x$$

This formula allows us to find the rate of change of the hyperbolic function coth(x) with respect to the variable 'x'.

## How do you prove the derivative of coth x?

There are various methods to derive derivatives of coth x. Three common methods are;

1. First Principle
2. Chain Rule
3. Quotient Rule

Each method provides a different way to compute the differentiation of cothx. By using these methods, we can mathematically prove the formula for finding the coth x derivative.

## Derivative of coth x by first principle

According to the first principle of derivative, the ln cothx derivative is equal to -cosh^2x. The derivative of a function by first principle refers to finding a general expression for the slope of a curve by using algebra. It is also known as the delta method. The derivative is a measure of the instantaneous rate of change, which is equal to,

$$f'(x) = \lim_{h\to 0} \frac{f(x+h)-f(x)}{h}$$

The technique of finding derivative of a function by first principle is known as derivative by definition

## Proof of derivative of coth x by first principle

To prove the derivative of coth x by using first principle, we start by replacing f(x) by coth x.

$$f′(x)=\lim_{h→0}\frac{f(x+h)−f(x)}{h}$$

$$f'(x) = \lim_{h\to 0} \frac{\coth(x+h)-\coth(x)}{h}$$

Since coth x = cosh x/sinh x, therefore,

$$f’(x)=\lim_{h\to 0}\frac{\frac{\cosh(x+h)}{\sinh(x+h)} –\frac{\cosh x}{\sinh x}}{h}$$

More simplification,

$$f’(x)=\lim_{h\to 0} \frac{\sinh x\cosh(x+h)–\sinh(x+h)\cosh x}{h\sinh x\sinh(x+h)}$$

Now, by the trigonometric formula, sinh A cosh B - cosh A sinh B = sinh (A - B)

$$f’(x)=\lim_{h\to 0}\frac{\sinh (x-(x+h))}{h\sinh x\sinh(x+h)}$$

$$f’(x)=\lim_{h\to 0}\frac{\sinh(-h)}{h\sinh x\sinh(x+h)}$$

We have sinh(-h) = -sinh(h), therefore,

$$f’(x)=-\lim_{h\to 0}\left(\frac{\sinh h}{h}\right)\times\left(\lim_{h\to 0}\frac{1}{\sinh x\sinh(x+h)}\right)$$

When h approaches to zero,

$$f’(x) = -1 \times\frac{1}{\sinh x}\times\sinh x$$

We can write it as;

$$f’(x)=-\DeclareMathOperator{\csch}{csch}\csch^2x$$

Since the function tanh x is a hyperbolic function. Therefore, the process of finding derivatives of a hyperbolic function is known as hyperbolic differentiation.

## Derivative of coth x by chain rule

To calculate the derivative of coth x, we can use the chain rule since the coth x function can be expressed as a combination of two functions. The chain rule of derivatives states that the derivative of a composite function is equal to the derivative of the outer function multiplied by the derivative of the inner function. The derivative chain rule is defined as;

$$\frac{dy}{dx}=\frac{dy}{du} × \frac{du}{dx}$$

## Proof of derivative of coth x by chain rule

To prove the derivative of coth x by using chain rule, we start by assuming that,

$$f(x)=\coth x=\frac{1}{\tanh x}=(\tanh x)^{-1}$$

By using chain rule solver,

$$f’(x)=(-1)(\tanh x)^{-2}\frac{d}{dx}(\DeclareMathOperator{\sech}{sech}\sech^2x)$$

Simplifying,

$$f’(x)=-\frac{1}{\tanh^2x}\times(\sech^2x)$$

Again,

$$f’(x)=-\frac{\sech^2x}{\tanh^2x}$$

Since sinh x / cosh x =tanh x and 1/cosh x = sec x, therefore we have

$$f’(x)=-\DeclareMathOperator{\csch}{csch}\csch^2x$$

## Derivative of coth x using quotient rule

Since the cotangent is the reciprocal of the tangent. Therefore, the derivative of cot can also be calculated by using the quotient rule. The quotient rule is defined as;

$$(\frac{u}{v})'=\frac{u'v-uv'}{v^2}$$

## Proof of derivative of coth x by quotient rule

To prove the derivative of coth x, we can write it,

$$f(x)=\coth x=\frac{1}{\tanh x}=\frac{u}{v}$$

Supposing that u = 1 and v = tanh x. Now by quotient rule,

$$(\frac{u}{v})'=\frac{u'v-uv'}{v^2}$$

$$f'(x)=\frac{\tanh x\frac{d}{dx}(1) – 1.\frac{d}{dx}(\tanh x)}{(\tanh x)^2}$$

$$f'(x)=\frac{\tanh x(0) - 1 (\DeclareMathOperator{\sech}{sech}\sech^2x)}{\tanh^2x}$$

$$f'(x)=-\frac{\sech^2x}{\tanh^2x}$$

$$f'(x)=-\DeclareMathOperator{\csch}{csch}\csch^2x$$

Hence, we have derived the derivative of coth x using the quotient rule of differentiation calculator.

## How to find the derivative of coth x with a calculator?

The easiest way to calculate the derivative of coth x is by using an online tool. You can use our derivative calculator for this. Here, we provide you a step-by-step way to calculate derivatives by using this tool.

1. Write the function as coth x in the “enter function” box. In this step, you need to provide input value as a function as you have to calculate the derivative of coth x.
2. Now, select the variable by which you want to differentiate coth x. Here you have to choose ‘x’.
3. Select how many times you want to differentiate cotangent x. In this step, you can choose 2 for second, 3 for third derivative and so on.
4. Click on the calculate button. After this step, the differential calculator will provide you a step-by-step solution.