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;

1. Quotient Rule
2. 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.

1. 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.
2. Now, select the variable by which you want to differentiate sec x.tan x. Here you have to choose ‘x’.
3. 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.
4. Click on the calculate button. After this step, you will get the derivative of sech x.tanh x within a few seconds.