Derivative of sech x

Learn what is the derivative of sech x with easy and step-wise proof. Also understand to prove the derivative of secant by chain rule and quotient rule.

Alan Walker-

Published on 2023-05-26

Introduction to the Derivative of sech x?

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

What is the derivative of sechx?

The derivative of sech(x) is defined as the negative of sech x multiplied by tanh x and denoted by d/dx(sech x). This derivative represents the rate of change of the hyperbolic function sech x with respect to the variable 'x'. The function sech x is composed of two exponential functions, e^x and e^-x, as defined by the equation;

$\sech x =\frac{1}{\cosh x}=\frac{2}{e^x+e^{-x}}$

By finding the derivative of sech x, we can gain a better understanding of the behavior of this function in calculus and other areas of mathematics.

Sechx differentiation formula

The derivative of the hyperbolic secant function, denoted by d/dx(sech x), can be calculated using the formula -sech x tanh x. Mathematically, 

$\frac{d}{dx}(\sech x)=-\sech x\tanh x$

This formula represents the product of the secant and tangent functions, and it provides a way to determine the rate of change of sech x with respect to the variable 'x'. Specifically, the d/dx sechx is equal to -sech x times tanh x. 

How do you prove the derivative of sechx?

 There are multiple ways to derive derivatives of sech x. Three commonly used methods are;

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

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

Derivative of sech x by first principle

According to the first principle of derivative, the ln sech x derivative is equal to -sech xtanh 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,

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

You can also use our derivative by definition calculator as it also follows the above formula. 

Proof of sechx derivative by first principle

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

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

Since $\sech x=\frac{1}{\cosh x}$, therefore, 

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

More simplifying, 

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

By sum to product formulas, cosh A - cosh B = 2 sin (A+B)/2 sin (A-B)/2. So,

$f'(x)=\frac{1}{\cosh x}\lim_{h\to 0} \frac{1}{h}\left(\frac{2\sinh\frac{x+x+h}{2} \sinh\frac{x-x-h}{2}}{\cosh(x + h)}\right)$

$f'(x)=\frac{1}{\cosh x}\lim_{h\to 0}\frac{1}{h}\left(\frac{2\sinh\frac{2x+h}{2} \sinh\frac{-h}{2}}{\cosh(x + h)}\right)$

Multiply and divide by h/2,

$f'(x)=-\frac{1}{\cosh x}\lim_{h\to 0}(\frac{1}{h})(\frac{h}{2})\left(\frac{2\sinh\frac{2x+h}{2}\frac{\sinh h/2}{h/2}}{\cosh(x + h)}\right)$

When h approaches to zero, h/2 also approaches to zero, therefore, 

$f'(x)=-\frac{1}{\cosh x}\lim_{h\to 0}\frac{\sinh(h/2)}{(h/2)}\times\lim_{h\to 0}\frac{\sinh\frac{2x+h}{2}}{\cosh(x + h)}$

We know that lim sinh x/x=1, 

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

We know that 1/cosh x = sech x and sinh x/cosh x = tanh x. So

$f'(x)=-\sech x\tanh x$

Hence the derivative of sechx is equal to the product of sech x and tanh x. The process of finding differentiation of sech x is called hyperbolic differentiation.

Differentiation of sechx by chain rule

To calculate the sechx differentiation, we can use the chain rule since the cosine 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 chain rule of derivative is defined as;


Proof of sechx derivative by chain rule 

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

$f(x)=\sech x=\frac{1}{\cosh x}=(\cosh x)^{-1}$

By using the leibniz notation calculator,

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


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


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

Since sinh x / cosh x =tan x and 1/cosh x = sech x, therefore we have

$f'(x)=\sech x\tanh x$

Derivative of sech x using quotient rule

Another method for finding the derivative of sech(x) 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 formula for the quotient rule derivative calculator is defined as:


Proof of sech x derivative by quotient rule 

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

$f(x)=\sech x=\frac{1}{\cosh x}=\frac{u}{v}$

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


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

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

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

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

$f'(x)=-\sech x\tanh x$

How to find the sechx derivative with a calculator?

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

  1. Write the function as sech 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 sechx.
  2. Now, select the variable by which you want to differentiate sech x. Here you have to choose ‘x’.
  3. Select how many times you want to differentiate secant x. In this step, you can choose 2 for second, 3 for third derivative and so on.
  4. Click on the calculate button.

After completing these steps, you will receive the sech 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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