Differentiation Of Hyperbolic Functions - Free Mathematics Tutorials

Differentiation of Hyperbolic Functions

Formulas and examples, with detailed solutions, on the derivatives of hyperbolic functions are presented. For definitions and graphs of hyperbolic functions go to Graphs of Hyperbolic Functions

Table of Hyperbolic Functions and Their Derivatives

function	derivative
\( f(x) = \sinh x \)	\( f '(x) = \cosh x \)
\( f(x) = \cosh x \)	\( f '(x) = \sinh x \)
\( f(x) = \tanh x = \dfrac{\sinh x}{\cosh x} \)	\( f '(x) = \operatorname{sech}^2 x \)
\( f(x) = \coth x = \dfrac{1}{\tanh x} = \dfrac{\cosh x}{\sinh x} \)	\( f '(x) = - \operatorname{csch}^2 x \)
\( f(x) = \operatorname{csch} x = \dfrac{1}{\sinh x} \)	\( f '(x) = - \operatorname{csch} x \coth x \)
\( f(x) = \operatorname{sech} x = \dfrac{1}{\cosh x} \)	\( f '(x) = - \operatorname{sech} x \tanh x \)

Examples with Solutions

Example 1

Find the derivative of \( f(x) = \sinh (x^2) \) Solution to Example 1: Let \( u = x^2 \) and \( y = \sinh u \) and use the chain rule to find the derivative of the given function \( f \) as follows. \( f '(x) = \dfrac{dy}{du} \dfrac{du}{dx} \) \( \dfrac{dy}{du} = \cosh u \), see formula above, and \( \dfrac{du}{dx} = 2x \) \( f '(x) = 2x \cosh u = 2x \cosh (x^2) \) Substitute \( u = x^2 \) in \( f '(x) \) to obtain \[ f '(x) = 2x \cosh (x^2) \]

Example 2

Find the derivative of \( f(x) = 2 \sinh x + 4 \cosh x \) Solution to Example 2: Let \( g(x) = 2 \sinh x \) and \( h(x) = 4 \cosh x \), function \( f \) is the sum of functions \( g \) and \( h \): \[ f(x) = g(x) + h(x) \]. Use the sum rule, \( f '(x) = g '(x) + h '(x) \), to find the derivative of function \( f \) \[ f '(x) = 2 \cosh x + 4 \sinh x \]

Example 3

Find the derivative of \( f(x) = \dfrac{\cosh x}{\sinh (x^2)} \) Solution to Example 3: Let \( g(x) = \cosh x \) and \( h(x) = \sinh x^2 \), function \( f \) is the quotient of functions \( g \) and \( h \): \( f(x) = \dfrac{g(x)}{h(x)} \). Hence we use the quotient rule , \[ f '(x) = \dfrac{h(x) g '(x) - g(x) h '(x)}{h(x)^2} \] to find the derivative of function \( f \). \( g '(x) = \sinh x \) \( h '(x) = 2x \cosh x^2 \) (see example 2 above) \[ f '(x) = \dfrac{( \sinh x^2 ) ( \sinh x ) - ( \cosh x ) ( 2x \cosh x^2 )}{( \sinh x^2 )^2} \]

Example 4

Find the derivative of \( f(x) = (\sinh x)^2 \) Solution to Example 4: Let \( u = \sinh x \) and \( y = u^2 \), Use the chain rule to find the derivative of function \( f \) as follows. \( f '(x) = \dfrac{dy}{du} \dfrac{du}{dx} \) \( \dfrac{dy}{du} = 2u \) and \( \dfrac{du}{dx} = \cosh x \) \( f '(x) = 2u \cosh x \) Put \( u = \sinh x \) in \( f '(x) \) obtained above \[ f '(x) = 2 \sinh x \cosh x \]

Exercises

Find the derivative of each function. 1 - \( f(x) = \sinh (x^3) \) 2 - \( g(x) = - \sinh x + 4 \cosh (x + 2) \) 3 - \( h(x) = \dfrac{\cosh x^2}{\sinh x} \) 4 - \( j(x) = - (\cosh x)^2 \)

Solutions to the Above Exercises

1 - \( f '(x) = (3x^2) \cosh (x^3) \) 2 - \( g '(x) = - \cosh x + 4 \sinh (x + 2) \) 3 - \( h '(x) = \dfrac{(2 x \sinh x^2)(\sinh x) - (\cosh x^2)(\cosh x)}{(\sinh x)^2} \) 4 - \( j '(x) = - 2 (\cosh x)(\sinh x) \)

More References and links

differentiation and derivatives Graphs of Hyperbolic Functions Rules of Differentiation of Functions in Calculus