Derivative of Hyperbolic Sine Function/Proof 1

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Theorem

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


Proof

\(\displaystyle D_x \left({ \sinh x }\right)\) \(=\) \(\displaystyle D_x \left({\dfrac {e^x - e ^{-x} } 2}\right)\) 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

$\blacksquare$