Derivative of Exponential Function/Proof 3

Theorem

Let $\exp$ be the exponential function.

Then:

$\map {\dfrac \d {\d x} } {\exp x} = \exp x$


Proof

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

$\blacksquare$


Sources

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