Test

$$\begin{align}

\frac{d^2}{dx^2} \ln f(x) &= \frac{f''(x)}{f(x)} - \left(\frac{f'(x)}{f(x)} \right)^2 \\[1ex] \frac{d^3}{dx^3} \ln f(x) &= \frac{f^{(3)}(x)}{f(x)} - 3 \frac{f'(x) f''(x)}{f(x)^2} + 2 \left(\frac{f'(x)}{f(x)} \right)^3 \\[1ex] \frac{d^4}{dx^4} \ln f(x) &= \frac{f^{(4)}(x)}{f(x)} - 4 \frac{f'(x) f^{(3)}(x)}{f(x)^2} - 3 \left(\frac{f(x)}{f(x)}\right)^2 + 12 \frac{f'(x)^2 f(x)}{f(x)^3} - 6 \left(\frac{f'(x)}{f(x)} \right)^4

\end{align}$$