Derivative of Exponential Function/Proof 3
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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
- For a video presentation of the contents of this page, visit the Khan Academy.