Derivative of Hyperbolic Sine/Proof 1

Theorem

$\map {\dfrac \d {\d x} } {\sinh x} = \cosh x$

Proof

\(\ds \map {\frac \d {\d x} } {\sinh x}\)

\(=\)

\(\ds \map {\frac \d {\d x} } {\dfrac {e^x - e ^{-x} } 2}\)

Definition of Hyperbolic Sine

\(\ds \)

\(=\)

\(\ds \frac 1 2 \paren {\map {\frac \d {\d x} } {e^x} - \map {\frac \d {\d x} } {e^{-x} } }\)

Linear Combination of Derivatives

\(\ds \)

\(=\)

\(\ds \frac 1 2 \paren {e^x - \paren {-e^{-x} } }\)

Derivative of Exponential Function, Chain Rule for Derivatives

\(\ds \)

\(=\)

\(\ds \frac {e^x + e^{-x} } 2\)

simplification

\(\ds \)

\(=\)

\(\ds \cosh x\)

Definition of Hyperbolic Cosine

$\blacksquare$