Math Doubts

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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