Derivative of sinh x

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

Alan Walker-

Published on 2023-05-26

What is the derivative of sinhx?

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

What is the derivative of sinh x?

The derivative of sinh(x), denoted as d/dx sinh x, represents the rate of change of the hyperbolic function sinh x with respect to the variable x.

By definition, sinh x can be expressed as (e^x - e^-x)/2, where e is the mathematical constant equal to approximately 2.718. The differentiation of sinh x is cosh x, which is also a hyperbolic function. Therefore, the sinh x differentiation can be written as:

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

Differentiation of sinhx formula

The formula to calculate the derivative of sinh x is equal to the cosh x function. Mathematically, it can be written as;

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

This formula is used to find the rate of change of the hyperbolic sine function with respect to the variable x. 

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

 There are different methods to derive derivatives of sinh x. Three common methods are;

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

Each method provides a different way to compute the e^x^2 derivative. By using these methods, we can mathematically prove the formula for finding the sinhx differentiation.

Derivative of sinh(x) by first principle

The derivative first principle says that the sinh derivative is equal to cosh 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.

Proof of sinhx derivative by first principle

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

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

Now, by using the trigonometric formula

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


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


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

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

$f'(x)=\cosh x$

Hence the sinh x derivative is equal to cosh x.

Derivative of sinh x by chain rule

To calculate the derivative of sinhx, we can use the chain rule since sinh x 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 chain rule of derivative is defined as;


The above rule can be used for hyperbolic differentiation

Proof of sinhx differentiation by chain rule 

To prove the derivative of sinh(x) by using chain rule, we will use the following hyperbolic formulas.

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

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

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

That is;

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

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


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


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

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

Derivative of sinh x using quotient rule

Since the sinh x is the ratio of two trigonometric ratios sine and cosine. Therefore, the derivative of sinhx can also be calculated by using the quotient rule. The quotient rule is defined as;


You can also use the quotient rule calculator with steps to verify the derivative of any hyperbolic function. 

Proof of differentiation of sinhx by quotient rule 

 To prove the derivative of hyperbolic sine, we can write it as,

$\frac{d}{dx}(\sinh x) =\frac{d}{dx}\left(\frac{1}{\DeclareMathOperator{\csch}{csch}\csch x}\right)$

By using quotient differentiation rule

$\frac{d}{dx}(\sinh x) =\frac{1.(\csch x\coth x)-\csch x(0)}{(\csch x )^2}$

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


$\coth x =\frac{\cosh x}{\sinh x}\quad\text{and}\quad \frac{1}{\csch x}=\sinh x$


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

Hence, we have derived the sinhx differentiation using the quotient rule of differentiation.

How to find the sinh x derivative with a calculator?

The easiest way to calculate the sinh derivativeis 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 sinh 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 sinh x.
  2. Now, select the variable by which you want to differentiate sinh x. Here you have to choose ‘x’.
  3. Select how many times you want to differentiate cosine 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, you will get the derivative of sinhx within a few seconds.

After completing these steps, you will receive the sinh 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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