Derivative of Exponential Function/Corollary 2

Corollary

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

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

Then:

$D_x \left({a^x}\right) = a^x \ln a$

Proof

From the definition of Power to a Real Number:

$a^x = e^{x \ln a}$

Thus from Corollary 1:

$D_x \left({a^x}\right) = D_x \left({e^{x \ln a}}\right) = \ln a e^{x \ln a} = a^x \ln a$

$\blacksquare$

Sources

• Murray R. Spiegel: Mathematical Handbook of Formulas and Tables (1968): $13.28$
• K.G. Binmore: Mathematical Analysis: A Straightforward Approach (1977)... (previous)... (next): $\S 14.7 \ (2)$