# Derivative of Hyperbolic Tangent function

## Formula

$\dfrac{d}{dx}{\, \tanh{x}}$ $\,=\,$ $\operatorname{sech^2}{x}$

### Introduction

Let $x$ represents a variable, the hyperbolic tangent function is written as $\tanh{x}$ in mathematics. The derivative of the hyperbolic tan function with respect to $x$ is written as follows.

$\dfrac{d}{dx}{\, \tanh{(x)}}$ $\,=\,$ $\operatorname{sech^2}{(x)}$

It is simply written in mathematical form as $(\tanh{x})’$ in differential calculus.

The differentiation of the hyperbolic tan function is equal to the square of hyperbolic secant function.

$\implies$ $\dfrac{d}{dx}{\, \tanh{x}}$ $\,=\,$ $\operatorname{sech^2}{x}$

#### Other forms

The derivative of hyperbolic tangent can be written in terms of any variable in mathematics.

#### Example

$(1) \,\,\,$ $\dfrac{d}{dp}{\, \tanh{(p)}}$ $\,=\,$ $\operatorname{sech^2}{(p)}$

$(2) \,\,\,$ $\dfrac{d}{dv}{\, \tanh{(v)}}$ $\,=\,$ $\operatorname{sech^2}{(v)}$

$(3) \,\,\,$ $\dfrac{d}{dy}{\, \tanh{(y)}}$ $\,=\,$ $\operatorname{sech^2}{(y)}$

### Proof

Learn how to derive the differentiation of hyperbolic tangent in differential calculus by the first principle of differentiation.

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Jun 26, 2023
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