# Derivative of Hyperbolic Cosecant function

## Formula

$\dfrac{d}{dx}{\, \operatorname{csch}{x}}$ $\,=\,$ $-\operatorname{csch}{x}\coth{x}$

### Introduction

The hyperbolic cosecant function is written as $\operatorname{csch}{x}$ in mathematical form, when $x$ represents a variable. The derivative of the hyperbolic cosecant function with respect to $x$ is written in the following mathematical form in differential calculus.

$\dfrac{d}{dx}{\, \operatorname{csch}{x}}$

The differentiation rule of the hyperbolic cosecant function is written simply as $(\operatorname{csch}{x})’$ in calculus. The differentiation of the hyperbolic cosecant function is equal to the negative sign of product of hyperbolic cosecant and cotangent functions.

$\implies$ $\dfrac{d}{dx}{\, \operatorname{csch}{x}}$ $\,=\,$ $-\operatorname{csch}{x}\coth{x}$

#### Other forms

The derivative of hyperbolic cosecant function can also be written in terms of any variable in mathematics.

#### Example

$(1) \,\,\,$ $\dfrac{d}{du}{\, \operatorname{csch}{(u)}}$ $\,=\,$ $-\operatorname{csch}{(u)}\coth{(u)}$

$(2) \,\,\,$ $\dfrac{d}{dt}{\, \operatorname{csch}{(t)}}$ $\,=\,$ $-\operatorname{csch}{(t)}\coth{(t)}$

$(3) \,\,\,$ $\dfrac{d}{dz}{\, \operatorname{csch}{(z)}}$ $\,=\,$ $-\operatorname{csch}{(z)}\coth{(z)}$

### Proof

Learn how to prove the differentiation of hyperbolic cosecant in differential calculus from the first principle of differentiation.

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