Derivative of General Exponential Function/Proof 3

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

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

• 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text {II}$. Calculus: Differentiation: Variable Index: Example $1$.