Derivative of Exponential Function/Proof 2

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Theorem

Let $\exp$ be the exponential function.

Then:

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


Proof

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

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

$\blacksquare$


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