Derivative of csch x

Learn what is the derivative of csch x with formula and proof. Also understand how to prove the derivative of csc by using quotient and product rule.

Alan Walker-

Published on 2023-05-26

Introduction to the Derivative of csch x?

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

What is the derivative of cschx?

The derivative of csch x with respect to the variable ‘x’ is equal to csch x .cot x. It is denoted by d/dx(csch x). It is the rate of change of the hyperbolic function cos s. sech x. By definition, the hyperbolic function sech x consists of two exponential functions, e^x and e^-x such that:

$csc x =\frac{2}{e^x-e^{-x}}$

Differentiation of cschx formula

The formula of derivative of csch(x) is equal to the negative product of the cotangent and csch xant function, that is;

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

How do you prove the differential of csch x?

 There are numerous ways to derive the derivatives of csch x. Therefore, we can prove the derivative cschx by using;

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

Derivative of csch(x) by first principle

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 derivative definition calculator also uses the above limit definition of derivatives. 

Proof of derivative of csch x by first principle

To prove the derivative of cschx by using first principle, replace f(x) by csch x in,

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

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

Since $\csch x=\frac{1}{\sin x}$, therefore, 

$f'(x) = \lim_{h\to 0} \frac{\frac{1}{\sinh(x+h)}-\frac{1}{\sinh x}}{h}$

More simplification, 

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


$f'(x)=\lim_{h\to 0} \frac{\sinh x –\sinh (x+h)}{h}\times\lim\frac{1}{\sinh x\sinh(x+h)}$

When h approaches zero, the left limit will be equal to the derivative of sinh x.

$f’(x)=-\cosh x\cdot\frac{1}{\sinh^2x}$

We can write it as;

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

Derivative of csch x by chain rule

The differentiation of csch x can be calculated by using chain rule because the cosine function can be written as the combination of two functions. The chain rule of derivatives is defined as;

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

Proof of derivative of csch(x) by chain rule 

To prove the differentiation of csch(x) by using chain rule, assume that,

$f(x)=\DeclareMathOperator{\csch}{csch}\csch x=\frac{1}{\sinh x}=(\sinh x)^{-1}$

By using chain rule of differentiation,

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


$f’(x) = \frac{-1}{\sinh^2x}\cdot(\cosh x)$


$f’(x)=-\frac{\cosh x}{\sinh^2x}=-\frac{\cosh x}{\sinh x}\cdot\frac{1}{\sinh x}$

Since sin x / cos x =tan x and 1/cos x = sec x, therefore,

$f’(x)=-\coth x\csch x$

You can also use a chain rule calculator that allows you to calculate the rate of change of any function in a smart and easy way.

Derivative of cschx using quotient rule

Since the secant is the reciprocal of cosine. Therefore, the derivative of csch(x) can also be calculated by using the quotient rule. The quotient rule for two functions u(x) and v(x) is defined as;


The above formula is used by the quotient rule derivative calculator. So, you can use it to find step-by-step derivatives of any function.

Proof of differentiation of csch x by quotient rule 

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

$f(x)=\DeclareMathOperator{\csch}{csch}\csch x=\frac{1}{\sinh x}=u/v$

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


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

$f'(x)= \frac{\sinh x (0)-1(\cosh x)}{\sinh^2x}$

$f'(x)=\frac{-\cosh x}{\sinh^2x}$

$f'(x)=-\coth x · \csch x$

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

How to find the derivative of csch(x) with a calculator?

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

  1. Write the function as csch 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 cschx.
  2. Now, select the variable by which you want to differentiate csch x. Here you have to choose ‘x’.
  3. Select how many times you want to differentiate csch xant x. In this step, you can choose 2 for second, 3 to find the third derivative and so on.
  4. Click on the calculate button. After this step, you will get the derivative of csch x within a few seconds.
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