# Derivative Rule of Inverse Secant function

## Formula

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

### Introduction

Let $x$ represents a variable and also a real number, the inverse secant function is written as $\sec^{-1}{(x)}$ or $\operatorname{arcsec}{(x)}$ in inverse trigonometry. In differential calculus, the derivative or differentiation of the secant inverse function with respect to $x$ is written in two mathematical forms as follows.

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

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

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

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

##### Alternative forms

The derivative of secant inverse function can be written in terms of any variable. The following are some examples to learn how to write the derivative rule of inverse secant function in differential calculus.

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

$(2) \,\,\,$ $\dfrac{d}{dr}{\,\sec^{-1}{r}} \,=\, \dfrac{1}{|r|\sqrt{r^2-1}}$

$(3) \,\,\,$ $\dfrac{d}{dy}{\,\sec^{-1}{y}} \,=\, \dfrac{1}{|y|\sqrt{y^2-1}}$

#### Proof

Learn how to derive the differentiation formula for the inverse secant function by first principle.

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Jun 26, 2023

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