Derivative of Hyperbolic Sine/Proof 2

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Theorem

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


Proof

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

$\blacksquare$

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