## What is the Derivative of sech xtanh x?

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

## What is the derivative of sech xtanh x?

The derivative of sech x.tanh x with respect to the variable ‘x’ is equal to sech x. It is denoted by d/dx (sech x.tanh x). It is the rate of change of the product of sec x and tan x.

## Derivative of sech x.tanh x formula

The formula of differentiation of sech x.tanh x is equal to the sech x, that is;

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

## How do you prove the derivative of sech x.tanh x?

There are numerous ways to derive derivatives of products of sech x and tanh x. Therefore, we can prove the derivative of sech xtanh x by using;

- Quotient Rule
- Product Rule

## Derivative of sech xtanh x using quotient rule

Since the function sine is the ratio of opposite to the hypotenuse of a triangle. Therefore, the derivative of sech x tanh x can also be calculated by using the quotient rule. The quotient rule is defined as;

$\frac{d}{dx}\left(\frac{f}{g}\right) = \frac{f(x)g'(x)-g(x)f'(x)}{(g(x))^2}$

## Proof of derivative of sech xtanh x by quotient rule

To prove the derivative of sech x tanh x, we can write it,

$f(x)=\DeclareMathOperator{\sech}{sech}\sech x\tanh x$

Since,

$\sech x=\frac{1}{\cosh x}$

And,

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

So,

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

Assume that,

$u=\sinh x$

$v=\cosh^2x$

Now by the quotient rule,

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

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

$f'(x)=\frac{\cosh^3x-(2\cosh x.\sinh^2x)}{\cosh^4x}$

We get,

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

Since,

$\sech^2x-\tanh^2x=1$

Therefore,

$f'(x)=\sech x$

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

## Derivative of square sech xtanh x by product rule

The derivative of the square root of x can be calculated by using product rule because an algebraic function can be written as the combination of two functions. The product rule of derivatives is defined as;

$\frac{d}{dx}[uv] = u.v'+u'.v$

The product rule derivative calculator also uses the above formula to provide you a quick and step-by-step solution.

## Proof of derivative of sech xtanh x by product rule

To prove the derivative of sech x tanhx by using product rule, assume that,

$f(x)=\DeclareMathOperator{\sech}{sech}\sech x\tanh x$

By using product rule of differentiation calculator,

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

We get,

$f'(x)=\sech^3x-\sech x.\tanh^2x$

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

Since,

$\sech^2x-\tanh^2x=1$

Hence,

$f'(x)=\sech x$

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

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

The easiest way to calculate the derivative of sech xtanh 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.

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