Third Derivative of Natural Logarithm Function

Theorem

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

Then:

$\map {D^3_x} {\ln x} = \dfrac 2 {x^3}$

Proof

\(\ds \map {D^3_x} {\ln x}\)

\(=\)

\(\ds \map {D_x} {\map {D^2_x} {\ln x} }\)

\(\ds \)

\(=\)

\(\ds \map {D_x} {-\dfrac 1 {x^2} }\)

Second Derivative of Natural Logarithm Function

\(\ds \)

\(=\)

\(\ds -\paren {-2 \paren {\dfrac 1 {x^3} } }\)

Power Rule for Derivatives: Integer Index

\(\ds \)

\(=\)

\(\ds \dfrac 2 {x^3}\)

$\blacksquare$

Sources

• 1963: Morris Tenenbaum and Harry Pollard: Ordinary Differential Equations ... (previous) ... (next): Chapter $1$: Basic Concepts: Lesson $3$: The Differential Equation