How to solve $n$-th derivative of logarithm function?

Question

How to slove the $n$-th derivative of logarithm function $$\frac{d^n}{dx^n} \ln \left(\frac{p}{ax+b}+\frac{1-p}{b}\right) $$ where $p \in [0,1]$, $a$ and $b$ are positive numbers

Answer 1

These are the first derivatives. Try to find a general formula for the $n-$th

$$ \begin{array}{r|r} \text{order} & \text{derivative}\\ \hline 1 & -\frac{a}{b+a x}+\frac{a-a p}{b-a (p-1) x}\\ 2 & \frac{a^2}{(b+a x)^2}-\frac{(a-a p)^2}{(b-a (p-1) x)^2} \\ 3 & 2\left(-\frac{ a^3}{(b+a x)^3} +\frac{ (a-a p)^3}{(b-a (p-1) x)^3}\right)\\ 4 & 6\left(\frac{ a^4}{(b+a x)^4}-\frac{ (a-a p)^4}{(b-a (p-1) x)^4} \right)\\ 5 & 24\left(-\frac{a^5}{(b+a x)^5} +\frac{(a-a p)^5}{(b-a (p-1) x)^5}\right)\\ 6 & 120 \left(\frac{a^6}{(b+a x)^6}-\frac{(a-a p)^6}{(b-a (p-1) x)^6}\right) \\ 7 & 720 \left(-\frac{a^7}{(b+a x)^7}+\frac{(a-a p)^7}{(b-a (p-1) x)^7}\right) \\ 8 & 5040 \left(\frac{a^8}{(b+a x)^8}-\frac{(a-a p)^8}{(b-a (p-1) x)^8}\right) \\ 9 & 40320 \left(-\frac{a^9}{(b+a x)^9}+\frac{(a-a p)^9}{(b-a (p-1) x)^9}\right) \\ 10 & 362880 \left(\frac{a^{10}}{(b+a x)^{10}}-\frac{(a-a p)^{10}}{(b-a (p-1) x)^{10}}\right) \\ 11 & 3628800 \left(-\frac{a^{11}}{(b+a x)^{11}}+\frac{(a-a p)^{11}}{(b-a (p-1) x)^{11}}\right) \\ 12 & 39916800 \left(\frac{a^{12}}{(b+a x)^{12}}-\frac{(a-a p)^{12}}{(b-a (p-1) x)^{12}}\right) \\ \end{array} $$

I my humble opinion it is

$$f^{(n)}(x)= (n-1)! \left[(-1)^n\left(\frac{a}{a x+b}\right)^n+(-1)^{n+1} \left(\frac{a(1-p)}{b+a (1-p) x}\right)^n\right]$$