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(Created page with "<math display="block">\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{a...") |
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