Derivative of General Exponential Function

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Corollary to Derivative of Exponential Function

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

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


Then:

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


Proof 1

\(\ds \map {\frac \d {\d x} } {a^x}\) \(=\) \(\ds \map {\frac \d {\d x} } {e^{x \ln a} }\) Definition of Power to Real Number
\(\ds \) \(=\) \(\ds \ln a e^{x \ln a}\) Derivative of Exponential of a x
\(\ds \) \(=\) \(\ds a^x \ln a\)

$\blacksquare$


Proof 2

\(\ds \lim_{h \mathop \to 0} \frac {a^{x + h} - a^x} h\) \(=\) \(\ds a^x \lim_{h \mathop \to 0} \frac {a^h - 1} h\) 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
\(\ds \) \(=\) \(\ds a^x \ln a\)

$\blacksquare$


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