Second Derivative of Natural Logarithm Function

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

Then:
 * $\map {\dfrac {\d^2} {\d x^2} } {\ln x} = -\dfrac 1 {x^2}$

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

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