Derivative of Exponential Function

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Theorem

Let $\exp$ be the exponential function.

Then:

$D_x \left({\exp x}\right) = \exp x$


Corollary 1

Let $c \in \R$.

Then:

$D_x \left({\exp \left({c x}\right)}\right) = c \exp \left({c x}\right)$


Corollary 2

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

\(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(\displaystyle D_x \left({\exp x}\right)\) \(=\) \(\displaystyle \) \(\displaystyle \lim_{h \to 0} \frac {\exp \left({x+h}\right) - \exp x} h\) \(\displaystyle \) \(\displaystyle \)          by definition of derivative          
\(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(=\) \(\displaystyle \) \(\displaystyle \lim_{h \to 0} \frac {\exp x \cdot \exp h - \exp x} h\) \(\displaystyle \) \(\displaystyle \)          Exponent of Sum          
\(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(=\) \(\displaystyle \) \(\displaystyle \lim_{h \to 0} \frac {\exp x \left({\exp h - 1}\right)} h\) \(\displaystyle \) \(\displaystyle \)                    
\(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(=\) \(\displaystyle \) \(\displaystyle \exp x \left({\lim_{h \to 0} \frac {\exp h - 1} h}\right)\) \(\displaystyle \) \(\displaystyle \)          Multiple Rule for Limits of Functions, as $\exp x$ is constant          
\(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(=\) \(\displaystyle \) \(\displaystyle \exp x\) \(\displaystyle \) \(\displaystyle \)          Derivative of Exponential at Zero          

$\blacksquare$


Proof 2

We use the fact that the exponential function is the inverse of the natural logarithm function:

$y = e^x \iff x = \ln y$
\(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(\displaystyle \dfrac {\mathrm d x} {\mathrm d y}\) \(=\) \(\displaystyle \) \(\displaystyle \dfrac 1 y\) \(\displaystyle \) \(\displaystyle \)          Derivative of Natural Logarithm Function          
\(\displaystyle \) \(\displaystyle \implies\) \(\displaystyle \) \(\displaystyle \dfrac {\mathrm d y} {\mathrm d x}\) \(=\) \(\displaystyle \) \(\displaystyle \dfrac {1} {1 / y}\) \(\displaystyle \) \(\displaystyle \)          Derivative of Inverse Function          
\(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(=\) \(\displaystyle \) \(\displaystyle y\) \(\displaystyle \) \(\displaystyle \)                    
\(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(=\) \(\displaystyle \) \(\displaystyle e^x\) \(\displaystyle \) \(\displaystyle \)                    

$\blacksquare$


Proof 3

\(\displaystyle \) \(\displaystyle \) \(\displaystyle \) \(\displaystyle D_x (\ln e^x)\) \(=\) \(\displaystyle \) \(\displaystyle D_x (x)\) \(\displaystyle \) \(\displaystyle \)          Exponential of Natural Logarithm          
\(\displaystyle \) \(\displaystyle \implies\) \(\displaystyle \) \(\displaystyle \frac{1}{e^x}D_x (e^x)\) \(=\) \(\displaystyle \) \(\displaystyle 1\) \(\displaystyle \) \(\displaystyle \)          Chain rule, Derivatives of Natural Log and Identity functions.          
\(\displaystyle \) \(\displaystyle \implies\) \(\displaystyle \) \(\displaystyle D_x (e^x)\) \(=\) \(\displaystyle \) \(\displaystyle e^x\) \(\displaystyle \) \(\displaystyle \)          multiply both sides by $e^x$          

$\blacksquare$


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