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
This 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

$u = \left|\dfrac {x}{\Delta x}\right|$

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

Substitute $u$ into the above equation, and since $u \to \infty$ we will set $u > 1$ for simplicity. Note that because the domain of the natural logarithm is $(0..\infty)$, $x$ is positive.

Proof 3
This proof assumes the definition of the natural logarithm as the inverse of the exponential function, as defined by a differential equation.

$y = \exp(x) \iff \ln y = x$,

The result follows from the definition of the antiderivative and the defined initial condition $(x_0,y_0)=(0,1)$.