Derivative of Exponential Function/Proof 3

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Theorem

Let $\exp$ be the exponential function.

Then:

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


Proof

\(\displaystyle \map {D_x} {\ln e^x}\) \(=\) \(\displaystyle \map {D_x} x\) Exponential of Natural Logarithm
\(\displaystyle \leadsto \ \ \) \(\displaystyle \frac 1 {e^x} \map {D_x} {e^x}\) \(=\) \(\displaystyle 1\) Chain Rule for Derivatives, Derivative of Natural Logarithm Function, Derivative of Identity Function
\(\displaystyle \leadsto \ \ \) \(\displaystyle \map {D_x} {e^x}\) \(=\) \(\displaystyle e^x\) multiply both sides by $e^x$

$\blacksquare$


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