Math Doubts

Differentiation of Hyperbolic sine

Formula

$\dfrac{d}{dx}{\, \sinh{x}} \,=\, \cosh{x}$

Introduction

In some cases, the hyperbolic sine often appears in differential calculus. Hence, it is essential to learn the derivative of hyperbolic sine function.

The hyperbolic sine function is written as $\sinh{x}$ mathematically if $x$ is used to represent a variable. The differentiation or derivative of $\sinh{x}$ function with respect to $x$ is written as $\dfrac{d}{dx}{\, \sinh{x}}$ in calculus and it is equal to $\cosh{x}$.

$\dfrac{d}{dx}{\, \sinh{x}}$ $\,=\,$ $\cosh{x}$

Other form

Remember, the variable can be denoted by any symbol by the derivative of hyperbolic sine function should be written in terms of the respective variable.

Example

$(1) \,\,\,$ $\dfrac{d}{dm}{\, \sinh{(m)}}$ $\,=\,$ $\cosh{(m)}$

$(2) \,\,\,$ $\dfrac{d}{dk}{\, \sinh{(k)}}$ $\,=\,$ $\cosh{(k)}$

$(3) \,\,\,$ $\dfrac{d}{dz}{\, \sinh{(z)}}$ $\,=\,$ $\cosh{(z)}$

The example clears that the differentiation of hyperbolic sine is equal to hyperbolic cosine.

Proof

It is your turn to learn how to derive the differentiation of hyperbolic sine function formula in differential calculus.



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