Derivative of Exponential Function/Proof 2

From ProofWiki
Jump to: navigation, search

Theorem

Let $\exp$ be the exponential function.

Then:

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


Proof

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

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

$\blacksquare$


Sources