Derivative of General Exponential Function/Proof 1

Corollary to Derivative of Exponential Function

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

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

$\blacksquare$


Sources

  • 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 14.7 \ (2)$
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