Derivative Rule of Inverse Hyperbolic Sine function

Formula

$\dfrac{d}{dx}{\, \sinh^{-1}{x}}$ $\,=\,$ $\dfrac{1}{\sqrt{x^2+1}}$

Introduction

The inverse hyperbolic sine function is written as $\sinh^{-1}{(x)}$ or $\operatorname{arcsinh}{(x)}$ in mathematics when the $x$ represents a variable. The derivative of the inverse hyperbolic sine function with respect to $x$ is written in the following mathematical forms.

$(1).\,\,\,$ $\dfrac{d}{dx}{\, (\sinh^{-1}{x})}$

$(2).\,\,\,$ $\dfrac{d}{dx}{\, (\operatorname{arcsinh}{x})}$

Mathematically, the derivative of the inverse hyperbolic sine function is simply written as $(\sinh^{-1}{x})’$ or $(\operatorname{arcsinh}{x})’$ in differential calculus.

The differentiation of the hyperbolic inverse sin function with respect to $x$ is equal to multiplicative inverse of square root of sum of $1$ and $x$ squared.

$\implies$ $\dfrac{d}{dx}{\, \sinh^{-1}{x}}$ $\,=\,$ $\dfrac{1}{\sqrt{x^2+1}}$

Other forms

In mathematics, the derivative of inverse hyperbolic sine function can also be written in terms of any variable.

Example

$(1) \,\,\,$ $\dfrac{d}{db}{\, \sinh^{-1}{b}}$ $\,=\,$ $\dfrac{1}{\sqrt{b^2+1}}$

$(2) \,\,\,$ $\dfrac{d}{dl}{\, \sinh^{-1}{l}}$ $\,=\,$ $\dfrac{1}{\sqrt{l^2+1}}$

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

Proof

Learn how to prove the differentiation formula of hyperbolic inverse sine function by the first principle of differentiation.

Math Questions

The math problems with solutions to learn how to solve a problem.

Learn solutions

Math Worksheets

The math worksheets with answers for your practice with examples.

Practice now

Math Videos

The math videos tutorials with visual graphics to learn every concept.

Watch now

Subscribe us

Get the latest math updates from the Math Doubts by subscribing us.