Derivative of Exponential Function/Corollary 2

Corollary to Derivative of Exponential Function

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

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

Then:

$\map {D_x} {a^x} = a^x \ln a$

Proof 1

 $\displaystyle \map {D_x} {a^x}$ $=$ $\displaystyle \map {D_x} {e^{x \ln a} }$ Definition of Power to Real Number $\displaystyle$ $=$ $\displaystyle \ln a e^{x \ln a}$ Derivative of Exponential Function: Corollary 1 $\displaystyle$ $=$ $\displaystyle a^x \ln a$

$\blacksquare$

Proof 2

 $\displaystyle \lim_{h \mathop \to 0} \frac {a^{x + h} - a^x} h$ $=$ $\displaystyle a^x \lim_{h \mathop \to 0} \frac {a^h - 1} h$ Exponent Combination Laws: Product of Powers $\displaystyle$ $=$ $\displaystyle a^x \lim_{h \mathop \to 0} \frac {\exp \left({h \ln a}\right) - 1} h$ Definition of Power to Real Number $\displaystyle$ $=$ $\displaystyle a^x \lim_{h \mathop \to 0} \left({ \frac {\exp \left({h \ln a}\right) - 1} {h \ln a} }\right) \left({\frac {h \ln a} h}\right)$ $\displaystyle$ $=$ $\displaystyle a^x \lim_{h \mathop \to 0} \left({\frac {h \ln a} h}\right)$ Derivative of Exponential at Zero $\displaystyle$ $=$ $\displaystyle a^x \ln a$

$\blacksquare$