# Derivative of cscx formula

## Formula

$\dfrac{d}{dx}{\, (\csc{x})} \,=\, -\csc{x}\cot{x}$

The derivative or differentiation of cosecant function with respect to a variable is equal to the negative the product of cosecant and cotangent functions. This derivative rule is read as the derivative of $\csc{x}$ function with respect to $x$ is equal to the minus $\csc{x}$ times $\cot{x}$.

### Introduction

Take, $x$ as a variable, then according to trigonometry, the cosecant function is written as $\csc{x}$ or $\operatorname{cosec}{x}$ in mathematical form. The derivative of the cosecant function with respect to $x$ is written as the following mathematical form.

$\dfrac{d}{dx}{\, (\csc{x})} \,\,\,$ or $\,\,\, \dfrac{d}{dx}{\, (\operatorname{cosec}{x})}$

In differential calculus, the differentiation of the $\csc{x}$ function with respect to $x$ can be written as $\dfrac{d{\,(\csc{x})}}{dx}$ and also expressed as ${(\csc{x})}’$ simply.

#### Other form

The differentiation of the cosecant function formula can be written in the form of any variable.

$(1) \,\,\,$ $\dfrac{d}{dr}{\, (\csc{r})} \,=\, -\csc{r}\cot{r}$

$(2) \,\,\,$ $\dfrac{d}{dt}{\, (\csc{t})} \,=\, -\csc{t}\cot{t}$

$(3) \,\,\,$ $\dfrac{d}{dy}{\, (\csc{y})} \,=\, -\csc{y}\cot{y}$

### Proof

Learn how to derive the derivative of the cosecant function from first principle in differential calculus.

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Jun 26, 2023
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