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Wize High School Grade 12 Calculus Textbook > Derivatives of Exponential (& Logarithmic) Functions

Hyperbolic Functions (Bonus)

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Hyperbolic functions

The hyperbolic functions sinh⁡x\sinh x, cosh⁡x\cosh x, and tanh⁡x\tanh x are defined using the exponential function exe^x:
  • sinh⁡x=ex−e−x2\displaystyle \sinh x=\frac{e^x-e^{-x}}{2}
  • cosh⁡x=ex+e−x2\displaystyle \cosh x=\frac{e^x+e^{-x}}{2}
  • tanh⁡=sinh⁡xcosh⁡x=ex−e−xex+e−x\displaystyle \tanh=\frac{\sinh x}{\cosh x}=\frac{e^x-e^{-x}}{e^x+e^{-x}}
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Practice: Hyperbolic Function
Prove the following.
a) cosh⁡2x−sinh⁡2x=1\cosh^2x-\sinh^2x=1
b) ddx(sinh⁡x)=cosh⁡x\displaystyle \frac{d}{dx}\left(\sinh x\right)=\cosh x
c) ddx(cosh⁡x)=sinh⁡x\displaystyle \frac{d}{dx}\left(\cosh x\right)=\sinh x
d) ddx(tanh⁡x)=1cosh⁡2x\displaystyle \frac{d}{dx}\left(\tanh x\right)=\frac{1}{\cosh^2x}