Derivative of Hyperbolic Sine/Proof 3

Theorem

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

Proof

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

\(=\)

\(\ds -i \map {\frac \d {\d x} } {\sin i x}\)

Hyperbolic Sine in terms of Sine

\(\ds \)

\(=\)

\(\ds \cos i x\)

Derivative of Sine Function

\(\ds \)

\(=\)

\(\ds \cosh x\)

Hyperbolic Cosine in terms of Cosine

$\blacksquare$