Derivative of Hyperbolic Sine Function/Proof 1

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Theorem

$\map {D_x} {\sinh x} = \cosh x$


Proof

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

$\blacksquare$