Derivative of Natural Logarithm Function/Proof 3

Theorem
Let $\ln x$ be the natural logarithm function.

Then:
 * $D_x \left({\ln x}\right) = \dfrac 1 x$

Proof
This proof assumes the definition of the natural logarithm as the inverse of the exponential function as defined by differential equation:


 * $y = \dfrac {\mathrm dy}{\mathrm dx}$


 * $y = e^x \iff \ln y = x$

The result follows from the definition of the antiderivative and the defined initial condition:
 * $\left({x_0, y_0}\right) = \left({0, 1}\right)$