Intoduction to the Derivative of tanh x?

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

What is the derivative of tanhx?

The derivative of tanh(x), or the hyperbolic tangent function, can be expressed as d/dx tanh x = sec²x. This formula represents the slope of the tangent line at the point of change in the function. In a triangle, the tangent is the ratio of the opposite to adjacent sides. Mathematically, the formula for the hyperbolic tangent function is written as;

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

Derivative of tanh x formula

The derivative of the hyperbolic tangent function, or tanh x, is equal to the square of the hyperbolic secant function, or sech x. Mathematically, this is expressed as; 

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

How do you prove the tanhx derivative?

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

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

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

Derivative of tanh by first principle

The derivative of a function by first principle refers to finding a general expression for the curved line slope 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 limit definition of derivative.

Proof of derivative of tanhx by first principle

To prove the derivative of tanh x by using first principle, replace f(x) by tanh x. 

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

Now, by using trigonometric ratio tanh x =sinh x/cosh x , so,

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

Simplifying,

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

Since, 

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

It is the expansion of the sinh x function. Therefore, we can write the above equation as;

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

Again simplifying to get,

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

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

Separating limits,

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

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

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

Evaluating limit,

$f'(x)=\frac{1}{\cosh(x+0)\cosh x}$

Since the reciprocal of hyperbolic function cosine is equal to secant. Therefore,

$f'(x)=\frac{1}{\cosh^2x}=\DeclareMathOperator{\sech}{sech}\sech^2x$

Derivative of tanx by chain rule

To calculate the derivative of csc (2x), 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 derivative is defined as;

$\frac{dy}{dx}=\frac{dy}{du}\times\frac{du}{dx}$

Proof of derivative of tanh x by chain rule 

To prove derivative of tangent x by using chain rule, consider that, 

$u=\coth x=\frac{1}{\tanh x}$

We have, 

$\frac{d}{dx}(\tanh x)=\frac{d}{dx}\left(\frac{1}{\coth x}\right)$

$\implies \frac{d}{dx}(\tanh x)=\frac{d}{dx}\left(\frac{1}{u}\right)$

By chain rule, 

$\implies \frac{d}{dx}(\tanh x)=\frac{d}{du}\left(\frac{1}{u}\right)\times\frac{du}{dx}$

Since 

$\frac{du}{dx}=\DeclareMathOperator{\csch}{csch}\csch^2x$

$\frac{d}{dx}(\tanh x) =-\frac{1}{u^2}\times\cosh^2x$

Now using the value of u,

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

We get,

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

Hence the derivative of tanh x is equal to the square of sech x. You can also use the chain rule calculator to evaluate derivatives easily.

Derivative of tanx using quotient rule

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

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

The quotient rule calculator also uses the above formula which provides you with easy and quick solutions.

Proof of derivative of tanx by quotient rule 

 We have,

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

Now by applying a derivative with respect to x on the above equation.

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

By using quotient rule, 

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

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

Since,

$\cosh^2x-\sinh^2x=1$

Therefore,

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

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

Derivative of tanh x using product rule

Another method to find the derivative e^x^2 is the product rule formula which is used in calculus to calculate the derivative of the product of two functions. Specifically, the product rule is used when you need to differentiate two functions that are multiplied together. The formula for the product rule derivative calculator is:

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

In this formula, u and v are functions of x, and du/dx and dv/dx are their respective derivatives with respect to x.

Proof of derivative of tanhx by product rule 

The tangent can be written as;

$\tanh x=\frac{\sinh x}{\cosh x}=\sinh x\times\frac{1}{\cosh x}$

And, 

$\tanh x =\sinh x\DeclareMathOperator{\sech}{sech}\sech x

Applying derivative with respect to x,

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

Applying product rule of derivative, 

$\frac{d}{dx}(\tanh x)=\sech x.(\cosh x)+\sinh x (-\sech x\tanh x )$

$\frac{d}{dx}(\tanh x)=\frac{1}{\cosh x}\times\cosh x-\sinh x\times\frac{1}{\cosh x}\times\frac{\sinh x}{\cosh x}$

More simplification,

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

Since,

$-\cosh^2x+\sinh^2x =1$

Therefore,

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

Hence the tanh derivative is always equal to the square of sech^2x.

How to find the derivative of tanh x with a calculator?

The most easy way to calculate the derivative of tanhx 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 this tool.

1. Write the function as 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 tanh x.
2. Now, select the variable by which you want to differentiate tanh x. Here you have to choose ‘x’.
3. Select how many times you want to differentiate tangent x. In this step, you can choose 2 for second and 3 to find the third derivative.

Click on the calculate button. After this step, the derivative calculator will provide you a quick and accurate derivative of hyperbolic tangent function.