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:

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

$\blacksquare$


Sources

  • 1963: Morris Tenenbaum and Harry Pollard: Ordinary Differential Equations ... (previous) ... (next): Chapter $1$: Basic Concepts: Lesson $3$: The Differential Equation
  • 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 14.1$
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