Derivative of cosh x

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

Alan Walker-

Published on 2023-05-26

Introduction to the derivative of coshx

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

What is the derivative of coshx?

The derivative of cosh x, one of the fundamental hyperbolic functions, with respect to the variable 'x' is given by sinh x. This rate of change is denoted by d/dx x, and is essential in many mathematical applications. The hyperbolic function cosh x is defined as the sum of two exponential functions, e^x and e^-x, divided by two. Therefore, it can be expressed as:

$\cosh x=\frac{e^x+e^{-x}}{2}$

Knowing the coshx derivative can be particularly useful in calculating the slope of the function at any point, or finding the critical points, inflection points, or intervals of increase and decrease. Use our first derivative calculator to simplify your calculations and get accurate results instantly.

Coshx differentiation formula

The derivative formula of cosh x is equivalent to the hyperbolic sine function, which can be expressed as:

$\frac{d}{dx}(\cosh x)=\sinh x$

Understanding this formula is essential in various fields of mathematics, such as calculus and differential equations.

How do you prove the derivative of cosh(x)?

There are different methods to derive derivative of coshx. Three commonly used methods are;

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

Each method provides a different way to compute the cosh derivative. By using these methods, we can mathematically prove the formula for finding the coshx derivative.

Derivative of cosh(x) by first principle

The derivative first principle says that the differentiation of coshx is equal to the negative of sin x. 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,

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

This formula allows us to determine the rate of change of a function at a specific point by using the limit definition of the derivative calculator.

Proof of derivative of coshx by first principle

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

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

Now, by using trigonometric formula,

$\cosh(x+h)=\cosh x\cosh h-\sinh x\sinh h$

$f'x=\lim_{h\to 0}\frac{\cosh x\cosh h-\sinh x\sinh h-\cosh x}{h}$


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

As h approaches zero, sinh h/h becomes 1. So,

$f'x=\sinh x$

Hence the derivative of cosh x is equal to sinh x.

Derivative of cosh(x) by chain rule

The derivative of tan 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;


The above rule can be used for hyperbolic differentiation

Proof of derivative of cosh x by chain rule 

To prove the coshx differentiation by using chain rule, we will use the following hyperbolic formulas.

  • $\cosh x=\frac{e^x+e^{-x}}{2}$
  • $\sinh x=\frac{e^x-e^{-x}}{2}$

Using the above hyperbolic identities, we can write the cosh x derivative such as;

$\frac{d}{dx}(\cosh x)=\frac{d}{dx}\left(\frac{e^x+e^{-x}}{2}\right)$

That is;

$\frac{d}{dx}(\cosh x)=\frac{e^x+(-1)e^{-x}}{2}$

$\frac{d}{dx}(\cosh x)=\frac{e^x-e^{-x}}{2}$


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


$\frac{d}{dx}(\cosh x)=\frac{d}{dx}\left(\frac{e^x-e^{-x}}{2}\right)=\sinh x$

Hence the chain rule calculator also proved that the differentiation of coshx is always sinh x.

Derivative of cosh x using quotient rule

Another method for finding the derivative of cosh derivative is using the quotient rule, which is a formula for finding the derivative of a quotient of two functions. The derivative of cosecant can also be calculated using the quotient rule. The quotient rule is defined as:


Proof of derivative of cosh(x) by quotient rule 

 To prove the derivative of cosine, we can start by writing it as,

$\frac{d}{dx}(\cosh x)=\frac{d}{dx}\left(\frac{1}{\DeclareMathOperator{\sech}{sech}\sech x}\right)$

By using quotient rule, 

$\frac{d}{dx}(\cosh x)=\frac{1.(\sech x\tanh x)-\sech x(0)}{(\sech x)^2}$

$\frac{d}{dx}(\cosh x)=\frac{\sech x\tanh x}{(\sech x)^2}$


$\tanh x=\frac{\sinh x}{\cosh x}\quad\text{and}\quad \frac{1}{\sech x}=\cosh x$


$\frac{d}{dx}(\cosh x)=\frac{\sinh x}{\cosh x}\times\cosh x =\sinh x$

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

How to find the differentiation of coshx with a calculator?

The easiest way to calculate the derivative of coshx 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 the differential calculator.

  1. Write the function as cosh x in the “enter function” box. In this step, you need to provide input value as a function as you have to calculate the coshx differentiation.
  2. Now, select the variable by which you want to differentiate coshx. Here you have to choose ‘x’.
  3. Select how many times you want to differentiate cosh. 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, you will get the derivative of cosine x within a few seconds.

After completing these steps, you will receive the cosh x differentiation within seconds. Using online tools can make it much easier and faster to calculate derivatives, especially for complex functions.

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