Derivatives Of Hyperbolic Functions
In this chapter, we will be discuss the derivatives of hyperbolic functions - $\sinh {x}$, $\cosh {x}$, $\tanh {x}$, $\csch {x}$, $\sech {x}$ and $\coth {x}$. If you aren't familiar with hyperbolic functions you may want to go through this chapter
I. Derivative of $\sinh x$:
$\dfrac{d}{dx} \sinh {x} $
$= \dfrac{d}{dx} \dfrac{e^x - e^{-x}}{2} $
$= \dfrac{1}{2} \left[\dfrac{d}{dx} e^x - \dfrac{d}{dx} e^{-x} \right] $
$= \dfrac{1}{2} [ e^x - (- e^{-x})] $
$= \dfrac{e^x + e^{-x}}{2} $
$= \cosh {x}$
II. Derivative of $\cosh x$
$\dfrac{d}{dx} \cosh {x} $
$= \dfrac{d}{dx} \dfrac{e^x + e^{-x}}{2} $
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$= \dfrac{1}{2} \left[\dfrac{d}{dx} e^x + \dfrac{d}{dx} e^{-x} \right] $
$= \dfrac{1}{2} [ e^x + (- e^{-x})] $
$= \dfrac{e^x - e^{-x}}{2} $
$= \sinh {x}$
III. Derivative of $\tanh x$
$\dfrac{d}{dx} \tanh {x}$
$= \dfrac{d}{dx} \left(\dfrac{\sinh x}{\cosh x}\right)$
$= \dfrac{\cosh x\dfrac{d}{dx} \sinh x -\sinh\dfrac{d}{dx}\cosh x}{(\cosh^2 x)}$
$=\dfrac{\cosh^2 x - \sinh^2 x}{\cosh^2 x}$
$=\dfrac{1}{\cosh^2 x} $
$= \sech^2 x$
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IV. Derivative of $\csch x$
$\dfrac{d}{dx} \csch x$
$= \dfrac{d}{dx} \left(\dfrac{1}{\sinh x}\right)$
$= \dfrac{\sinh x\dfrac{d}{dx}1 -1\dfrac{d}{dx}\sinh x}{(\sinh^2 x)}$
$=-\dfrac{\cosh x}{\sinh^2 x}$
$=- \dfrac{\cosh x}{\sinh x} \times \dfrac{1}{\sinh x}$
$= -\coth x \cdot \csch x$
V. Derivative of $\sech x$
$\dfrac{d}{dx} \sech x$
$= \dfrac{d}{dx} \left(\dfrac{1}{\cosh x}\right)$
$= \dfrac{\cosh x\dfrac{d}{dx}1 -1\dfrac{d}{dx}\cosh x}{(\cosh^2 x)}$
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$=-\dfrac{\sinh x}{\cosh^2 x}$
$=- \dfrac{\sinh x}{\cosh x} \times \dfrac{1}{\cosh x}$
$= -\tanh x \cdot \sech x$
VI. Derivative of $\coth x$
$\dfrac{d}{dx} \coth {x}$
$= \dfrac{d}{dx} \left(\dfrac{\cosh x}{\sinh x}\right)$
$= \dfrac{\sinh x\dfrac{d}{dx} \cosh x -\cosh\dfrac{d}{dx}\sinh x}{(\sinh^2 x)}$
$=\dfrac{\sinh^2 x - \cosh^2 x}{\sinh^2 x}$
$=- \dfrac{1}{\sinh^2 x} $
$= -\csch^2 x$