Derivative of Natural Logarithm Function/Proof 3

Theorem

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

Then:

$\map {\dfrac \d {\d x} } {\ln x} = \dfrac 1 x$

Proof

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

$y = \dfrac {\d y} {\d x}$

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

\(\ds \frac {\d y} {\d x}\)

\(=\)

\(\ds y\)

Definition of Exponential Function

\(\ds \int \frac 1 y \rd y\)

\(=\)

\(\ds \int \rd x\)

Solution to Separable Differential Equation

\(\ds \)

\(=\)

\(\ds x + C_0\)

Integral of Constant where that constant is $1$

\(\ds \)

\(=\)

\(\ds \ln y + C_0\)

Definition 2 of Natural Logarithm: $x = \ln y$

The result follows from the definition of the antiderivative and the defined initial condition:

$\tuple {x_0, y_0} = \tuple {0, 1}$

$\blacksquare$