Nth Derivative of Natural Logarithm

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Theorem

The $n$th derivative of $\map \ln x$ for $n \ge 1$ is:

$\dfrac {\d^n} {\d x^n} \ln x = \dfrac {\paren {n - 1}! \paren {-1}^{n - 1} } {x^n}$


Proof

Proof by induction:

For all $n \in \N_{>1}$, let $\map P n$ be the proposition:

$\dfrac {\d^n} {\d x^n} \ln x = \dfrac {\paren {n - 1}! \paren {-1}^{n - 1} } {x^n}$


Basis for the Induction

$\map P 1$ is true, as this just says:

$\dfrac \d {\d x} \ln x = \dfrac 1 x$

This follows by Derivative of Natural Logarithm Function

This is our basis for the induction.


Induction Hypothesis

Now we need to show that, if $\map P k$ is true, where $k \ge 1$, then it logically follows that $\map P {k + 1}$ is true.


So this is our induction hypothesis:

$\dfrac {\d^k} {\d x^k} \ln x = \dfrac {\paren {k - 1}! \paren {-1}^{k - 1} } {x^k}$

Then we need to show:

$\dfrac {\d^{k + 1} } {\d x^{k + 1} } \ln x = \dfrac {k! \paren {-1}^k} {x^{k + 1} }$


Induction Step

This is our induction step:

\(\ds \frac {\d^{k + 1} } {\d x^{k + 1} } \ln x\) \(=\) \(\ds \map {\frac \d {\d x} } {\frac {\d^k} {\d x^k} \ln x}\)
\(\ds \) \(=\) \(\ds \map {\frac \d {\d x} } {\paren {k - 1}! \paren {-1}^{k - 1} x^{-k} }\) Induction Hypothesis
\(\ds \) \(=\) \(\ds \paren {k - 1}! \paren {-1}^{k - 1} \map {\frac \d {\d x} } {x^{-k} }\) Derivative of Constant Multiple
\(\ds \) \(=\) \(\ds \paren {k - 1}! \paren {-1}^{k - 1} \paren {k x^{-k - 1} }\) Power Rule for Derivatives
\(\ds \) \(=\) \(\ds k! \paren {-1}^k x^{-\paren {k + 1} }\) Definition of Factorial
\(\ds \) \(=\) \(\ds \frac {k! \paren {-1}^k} {x^{k + 1} }\)


So $\map P k \implies \map P {k + 1}$ and the result follows by the Principle of Mathematical Induction.


Therefore:

$\dfrac {\d^n} {\d x^n} \ln x = \dfrac {\paren {n - 1}! \paren {-1}^{n - 1} } {x^n}$

$\blacksquare$