Second Derivative of Natural Logarithm Function

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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:

$\dfrac {\d^2} {\d x^2} \ln x = \dfrac \d {\d x} \dfrac 1 x = -\dfrac 1 {x^2}$

$\blacksquare$


Sources