Derivative of Exponential Function/Proof 3

From ProofWiki
Jump to navigation Jump to search

Theorem

Let $\exp$ be the exponential function.

Then:

$\map {D_x} {\exp x} = \exp x$


Proof

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

$\blacksquare$


Sources