Derivative of General Exponential Function/Proof 1

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

\(\ds \map {\frac \d {\d x} } {a^x}\)

\(=\)

\(\ds \map {\frac \d {\d x} } {e^{x \ln a} }\)

Definition of Power to Real Number

\(\ds \)

\(=\)

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

Derivative of Exponential of a x

\(\ds \)

\(=\)

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

$\blacksquare$

Sources

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