# Differentiation of Hyperbolic Tangent

## Formula

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

### Introduction

The hyperbolic tangent is often appeared in differential calculus. So, everyone who learns the differential calculus must have to learn the derivative of hyperbolic tangent function.

If $x$ is a variable, then the hyperbolic tangent is written as $\tanh{x}$ in mathematical form. The derivative (or) differentiation of $\tanh{x}$ function with respect to $x$ is expressed as $\dfrac{d}{dx}{\, \tanh{x}}$ in differential calculus and it is equal to hyperbolic secant squared of $x$.

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

### Other form

The derivative of hyperbolic tangent is expressed in terms of a variable and the variable can be denoted by any symbol. So, the differentiation of hyperbolic tangent function is written in terms of the respective variable.

#### Example

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

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

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

It clears that the derivative of hyperbolic tangent function formula can be written in terms of any variable.

### Proof

Learn how to derive the differentiation of hyperbolic tangent is equal to square of hyperbolic secant in differential calculus.

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