Derivative of Hyperbolic Sine Function

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Theorem

$D_x \left({\sinh x}\right) = \cosh x$

where $\sinh$ is the hyperbolic sine and $\cosh$ is the hyperbolic cosine.


Proof 1

\(\displaystyle D_x \left({ \sinh x }\right)\) \(=\) \(\displaystyle D_x \left({\dfrac {e^x - e ^{-x} } 2}\right)\) Definition of Hyperbolic Sine
\(\displaystyle \) \(=\) \(\displaystyle \frac {D_x \left({e^x - e^{-x} }\right)} 2\) Derivative of Constant Multiple
\(\displaystyle \) \(=\) \(\displaystyle \frac {D_x \left({e^x}\right) - D_x \left({ e^{-x} }\right)} 2\) Linear Combination of Derivatives
\(\displaystyle \) \(=\) \(\displaystyle \frac {D_x \left({e^x}\right) - D_{-x} \left({ e^{-x} }\right) D_x \left({-x}\right)} 2\) Chain Rule
\(\displaystyle \) \(=\) \(\displaystyle \frac {e^x - e^{-x} D_x \left({-x}\right)} 2\) Derivative of Exponential Function
\(\displaystyle \) \(=\) \(\displaystyle \frac {e^x - e^{-x} \left({-1}\right)} 2\) Derivative of Identity Function
\(\displaystyle \) \(=\) \(\displaystyle \frac {e^x + e^{-x} } 2\)
\(\displaystyle \) \(=\) \(\displaystyle \cosh x\) Definition of Hyperbolic Cosine

$\blacksquare$


Proof 2

\(\displaystyle D_x \left({\sinh x}\right)\) \(=\) \(\displaystyle \lim_{h \mathop \to 0} \frac {\sinh \left({x + h}\right) - \sinh x} h\)
\(\displaystyle \) \(=\) \(\displaystyle \lim_{h \mathop \to 0} \frac {2 \cosh \left({\frac {x + h + x} 2}\right) \sinh \left({\frac {x + h - x} 2}\right)} h\) Hyperbolic Sine minus Hyperbolic Sine
\(\displaystyle \) \(=\) \(\displaystyle \lim_{h \mathop \to 0} \frac {2 \cosh \left({x + \frac h 2}\right) \sinh \left({\frac h 2}\right)} h\)
\(\displaystyle \) \(=\) \(\displaystyle \lim_{h \mathop \to 0} \frac {\cosh \left({x + \frac h 2}\right) \sinh \left({\frac h 2}\right)} {\frac h 2}\)
\(\displaystyle \) \(=\) \(\displaystyle \lim_{2 d \mathop \to 0} \frac {\cosh \left({x + d}\right) \sinh \left({d}\right)} d\) where $d = \dfrac h 2$
\(\displaystyle \) \(=\) \(\displaystyle \lim_{d \mathop \to 0} \frac {\cosh \left({x + d}\right) \sinh \left({d}\right)} d\)
\(\displaystyle \) \(=\) \(\displaystyle \cosh x \lim_{d \mathop \to 0} \frac {\sinh \left({d}\right)} d\)
\(\displaystyle \) \(=\) \(\displaystyle \cosh x \lim_{d \mathop \to 0} \frac { e^d - e^{-d} } {2 d}\) Definition of Hyperbolic Sine
\(\displaystyle \) \(=\) \(\displaystyle \cosh x \lim_{d \mathop \to 0} \frac { e^{2 d} - 1 } {2 d e^d}\)
\(\displaystyle \) \(=\) \(\displaystyle \cosh x \lim_{d \mathop \to 0} \frac 1 {e^d} \frac { e^{2 d} - 1 } {2 d}\)
\(\displaystyle \) \(=\) \(\displaystyle \cosh x \lim_{d \mathop \to 0} \frac 1 {e^d} \lim_{d \mathop \to 0} \frac { e^{2 d} - 1 } {2 d}\)
\(\displaystyle \) \(=\) \(\displaystyle \cosh x \lim_{d \mathop \to 0} \frac 1 {e^d} \lim_{2 d \mathop \to 0} \frac { e^{2 d} - 1 } {2 d}\)
\(\displaystyle \) \(=\) \(\displaystyle \cosh x \lim_{d \mathop \to 0} \frac 1 {e^d}\) Derivative of Exponential at Zero
\(\displaystyle \) \(=\) \(\displaystyle \cosh x\)

$\blacksquare$


Proof 3

\(\displaystyle D_x \paren {\sinh x}\) \(=\) \(\displaystyle -i D_x \paren {\sin i x}\) Hyperbolic Sine in terms of Sine
\(\displaystyle \) \(=\) \(\displaystyle \cos i x\) Derivative of Sine Function
\(\displaystyle \) \(=\) \(\displaystyle \cosh x\) Hyperbolic Cosine in terms of Cosine

$\blacksquare$


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