Single Variable Calculus, Chapter 7, Review Exercises, Section Review Exercises, Problem 54

Determine $f^{(n)} (x)$ to the function $f(x) = \ln (2x)$


$
\begin{equation}
\begin{aligned}

f(x) =& \ln 2 + \ln x
\\
\\
f'(x) =& \frac{d}{dx} (\ln 2) + \frac{d}{dx} (\ln x)
\\
\\
f'(x) =& 0 + \frac{1}{x}
\\
\\
f'(x) =& \frac{1}{x} \text{ or } x^{-1}
\\
\\
f''(x) =& \frac{d}{dx} (x^{-1})
\\
\\
f''(x) =& -x^{-2} \text{ or } \frac{-1}{x^2}
\\
\\
f'''(x) =& - \frac{d}{dx} (x^{-2})
\\
\\
f'''(x) =& 2x^{-3} \text{ or } \frac{2}{x^3}
\\
\\
f^4 (x) =& 2 \frac{d}{dx} (x^{-3})
\\
\\
f^4 (x) =& -6x^{-4} \text{ or } \frac{-6}{x^4}
\\
\\
f^5(x) =& -6 \frac{d}{dx} (x^{-4})
\\
\\
f^5(x) =& 24 x^{-5} \text{ or } \frac{24}{x^5}

\end{aligned}
\end{equation}
$


After taking several derivatives, we observe that

$\displaystyle f^{(n)} (x) = \frac{(-1)^{n - 1} (n - 1) !}{x^n}$