Derivative of General Exponential Function/Proof 2

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Let $a \in \R: a > 0$.

Let $a^x$ be $a$ to the power of $x$.


$\map {\dfrac \d {\d x} } {a^x} = a^x \ln a$


\(\ds \lim_{h \mathop \to 0} \frac {a^{x + h} - a^x} h\) \(=\) \(\ds a^x \lim_{h \mathop \to 0} \frac {a^h - 1} h\) Product of Powers
\(\ds \) \(=\) \(\ds a^x \lim_{h \mathop \to 0} \frac {\map \exp {h \ln a} - 1} h\) Definition of Power to Real Number
\(\ds \) \(=\) \(\ds a^x \lim_{h \mathop \to 0} \paren {\frac {\map \exp {h \ln a} - 1} {h \ln a} } \paren {\frac {h \ln a} h}\)
\(\ds \) \(=\) \(\ds a^x \lim_{h \mathop \to 0} \paren {\frac {h \ln a} h}\) Derivative of Exponential at Zero
\(\ds \) \(=\) \(\ds a^x \ln a\)