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
- 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): $\S 14.1$