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Wize University Calculus 1 Textbook > Derivatives

Hyperbolic Functions and their Derivatives

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Hyperbolic Functions and their Derivatives

While Trig Functions are defined for the circle, Hyperbolic Functions are their partners defined for the hyperbola.



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Hyperbolic Function Definitions

sinh⁡x=ex−e−x2cosh⁡x=ex+e−x2tanh⁡x=sinh⁡xcosh⁡x=ex−e−xex+e−x\displaystyle \boxed{\sinh x=\frac{e^x-e^{-x}}{2} }\hspace{1cm} \boxed{\cosh x=\frac{e^x+e^{-x}}{2}} \hspace{1cm} \boxed{\tanh x=\frac{\sinh x}{\cosh x}=\frac{e^x-e^{-x}}{e^x+e^{-x}}}

cschx=1sinh⁡xsechx=1cosh⁡xcoth⁡x=cosh⁡xsinh⁡x=ex+e−xex−e−x\displaystyle \boxed{\csch x=\frac{1}{\sinh x}} \hspace{1.3cm} \boxed{\sech x=\frac{1}{\cosh x}} \hspace{1.3cm} \boxed{\coth x=\frac{\cosh x}{\sinh x}=\frac{e^x+e^{-x}}{e^x-e^{-x}}}


Derivatives of Hyperbolic Functions

ddx(sinh⁡x)=cosh⁡xddx(cosh⁡x)=sinh⁡xddx(tanh⁡x)=sech2x\displaystyle \boxed{\frac{d}{dx}(\sinh x)=\cosh x} \hspace{1cm} \boxed{\frac{d}{dx}(\cosh x)=\sinh x} \hspace{1cm} \boxed{\frac{d}{dx}(\tanh x)=\sech^2 x}

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Example: Hyperbolic Derivatives

Find the derivative of f(x)=sinh⁡(x)cosh⁡(x)f(x) = \sinh(x)\cosh(x)

ddx[sinh⁡(x)cosh⁡(x)]=ddx[sinh⁡(x)]cosh⁡(x)+sinh⁡(x)ddx[cosh⁡(x)]=cosh⁡2(x)+sinh⁡2(x)\begin{aligned} \mydd{\sinh(x)\cosh(x)}{x}[bt] =& \mydd{\sinh(x)}{x}[bt]\cosh(x) + \sinh(x) \mydd{\cosh(x)}{x}[bt] \\ =& \cosh^2(x) + \sinh^2(x) \end{aligned}
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Find the derivative of f(x)=ln⁡(sinh⁡(2x))f(x) = \ln(\sinh(2x))

Extra Practice
Derivatives: Hyperbolic Functions
Find the derivative of f(x)=ln⁡(sinh⁡(2x))f(x) = \ln(\sinh(2x))
Derivatives: Hyperbolic Functions
Q. Find the derivative of f(x)=ln⁡(sinh⁡(2x))f(x) = \ln(\sinh(2x))
Find the derivative of f(x)=sinh⁡(x2)cosh⁡(x)f(x) = \sinh(x^2)\cosh(x)
Hyperbolic Functions and Derivatives
Differentiate y=ln⁡(cosh⁡x)+12cosh⁡2xy=\ln(\cosh x)+\frac{1}{2\cosh^2 x} and simplify fully.
Example: Solve Cosh Equation
Q:\bf{Q:} Solve for xx in the equation cosh⁡x=135\cosh x=\dfrac{13}{5} assuming x>0x>0.
Hyperbolic Functions
Differentiate cosh⁡ln⁡x\cosh \ln x in two ways: by simplifying before taking the derivative, and by using the derivative of cosh⁡x\cosh x along with the chain rule.
Derivatives: Hyperbolic Functions
Find the derivative of f(x)=ln⁡(sinh⁡(2x))f(x) = \ln(\sinh(2x))
Find the derivative of f(x)=sinh⁡(x2)cosh⁡(x)f(x) = \sinh(x^2)\cosh(x)
Example: Solve Cosh Equation
Q:\bf{Q:} Solve for xx in the equation cosh⁡x=135\cosh x=\dfrac{13}{5} assuming x>0x>0.
Hyperbolic Functions and Derivatives
Differentiate y=ln⁡(cosh⁡x)+12cosh⁡2xy=\ln(\cosh x)+\dfrac{1}{2\cosh^2x} and simplify fully.
Find the derivative of f(x)=sinh⁡(x2)cosh⁡(x)f(x) = \sinh(x^2)\cosh(x)
Hyperbolic Functions
Prove the following identity:
cosh⁡2x−sinh⁡2x=1\cosh^2x-\sinh^2x=1
Find the derivative of f(x)=sinh⁡(x2)cosh⁡(x)f(x) = \sinh(x^2)\cosh(x)
Find the derivative of f(x)=ln⁡(sinh⁡(2x))f(x) = \ln(\sinh(2x))