Derivative of Exponential Function/Corollary 2

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Corollary to Derivative of Exponential Function

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 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
\(\displaystyle \) \(=\) \(\displaystyle a^x \ln a\)

$\blacksquare$


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