Derivative of Exponential Function/Proof 1

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Theorem

Let $\exp$ be the exponential function.

Then:

$\map {\dfrac \d {\d x} } {\exp x} = \exp x$


Proof

\(\ds \map {\frac \d {\d x} } {\exp x}\) \(=\) \(\ds \lim_{h \mathop \to 0} \frac {\map \exp {x + h} - \exp x} h\) Definition of Derivative
\(\ds \) \(=\) \(\ds \lim_{h \mathop \to 0} \frac {\exp x \cdot \exp h - \exp x} h\) Exponential of Sum
\(\ds \) \(=\) \(\ds \lim_{h \mathop \to 0} \frac {\exp x \paren {\exp h - 1} } h\)
\(\ds \) \(=\) \(\ds \exp x \paren {\lim_{h \mathop \to 0} \frac {\exp h - 1} h}\) Multiple Rule for Limits of Real Functions, as $\exp x$ is constant
\(\ds \) \(=\) \(\ds \exp x\) Derivative of Exponential at Zero

$\blacksquare$


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