# Derivative Rule of Inverse Cosecant function

## Formula

$\dfrac{d}{dx}{\,\csc^{-1}{x}} \,=\, \dfrac{-1}{|x|\sqrt{x^2-1}}$

### Introduction

The inverse cosecant function is written as $\csc^{-1}{(x)}$ or $\operatorname{arccsc}{(x)}$ in inverse trigonometry when a variable is denoted by $x$. In differential calculus, the differentiation or derivative of the cosecant inverse function with respect to $x$ is written in following two different mathematical forms.

$(1) \,\,\,$ $\dfrac{d}{dx}{\,\Big(\csc^{-1}{(x)}\Big)}$

$(2) \,\,\,$ $\dfrac{d}{dx}{\,\Big(\operatorname{arccsc}{(x)}\Big)}$

The derivative of the inverse cosecant function with respect to $x$ is equal to the negative reciprocal of product of modulus of $x$ and square root of the subtraction of one from $x$ squared.

$\implies$ $\dfrac{d}{dx}{\,\Big(\csc^{-1}{(x)}\Big)}$ $\,=\,$ $-\dfrac{1}{|x|\sqrt{x^2-1}}$

##### Alternative forms

The derivative formula of cosecant inverse function can also be written in any variable. The following are some understandable examples to know how to express the derivative rule of inverse cosecant function in calculus.

$(1) \,\,\,$ $\dfrac{d}{dy}{\,\csc^{-1}{y}} \,=\, -\dfrac{1}{|y|\sqrt{y^2-1}}$

$(2) \,\,\,$ $\dfrac{d}{dm}{\,\csc^{-1}{m}} \,=\, -\dfrac{1}{|m|\sqrt{m^2-1}}$

$(3) \,\,\,$ $\dfrac{d}{dz}{\,\csc^{-1}{z}} \,=\, -\dfrac{1}{|z|\sqrt{z^2-1}}$

#### Proof

Learn how to prove the differentiation law for the inverse co-secant function by first principle.

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