Second Derivative of Natural Logarithm Function

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

Then:
 * $D^2_x \left({\ln x}\right) = -\dfrac 1 {x^2}$

Proof
From Derivative of Natural Logarithm Function:
 * $D \ln x = \dfrac 1 x$

From the Power Rule for Derivatives: Integer Index:
 * $D^2 \ln x = D \dfrac 1 x = -\dfrac 1 {x^2}$