Derivative of Natural Logarithm Function

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

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

Proof 1
Follows directly from the definition of the natural logarithm function as the primitive of the reciprocal function.

Proof 2
The proof assumes the definition of the natural logarithm as the inverse of the exponential function, $e^x := \displaystyle \lim_{n \to \infty} \left({1 + \frac x n}\right)^n$.

Define $u$ as follows

$\dfrac {1}{u} = \dfrac {\Delta x}{x}$

$\dfrac {1}{\Delta x} = \dfrac {u}{x}$

Then $\Delta x \to 0 \iff (u \to \infty \lor u \to -\infty$).

Case 1: $u \to \infty$

Substituting $u$ into the above equations, we have