Derivative of General Exponential Function

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Theorem

Let $a \in \R_{>0}$.

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 $e^{a x}$
\(\ds \) \(=\) \(\ds a^x \ln a\)

$\blacksquare$


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
\(\ds \) \(=\) \(\ds a^x \ln a\)

$\blacksquare$


Proof 3

Let $y = a^x$.

Then:

\(\ds \ln y\) \(=\) \(\ds x \ln a\) Logarithm of Power
\(\ds \leadsto \ \ \) \(\ds \dfrac 1 y \dfrac {\d y} {\d x}\) \(=\) \(\ds \ln a\) Derivative of Identity Function: Corollary
\(\ds \leadsto \ \ \) \(\ds \dfrac 1 {a^x} \dfrac {\d y} {\d x}\) \(=\) \(\ds \ln a\)
\(\ds \leadsto \ \ \) \(\ds \dfrac {\d y} {\d x}\) \(=\) \(\ds a^x \ln a\)

$\blacksquare$


Sources

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