Looking around the forum, I can't find good references for these, although there are lots of references to these formulas, I think it would be good to have these together, so here they are:
\(
\begin{tabular}{rl}
\frac{\partial}{\partial a} ({}^{x}{a}) & = \frac{1}{a} \sum_{k=1}^{x} \ln(a)^{k-1} \prod_{j=0}^{k} {}^{{(x-j)}}{a} \\
\frac{\partial}{\partial x} ({}^{x}{a}) & = T(a) \ln(a)^x \prod_{k=1}^{x} {}^{k}{a}
\end{tabular}
\)
where
\( T(a) = \left[\frac{\partial}{\partial x} ({}^{x}{a})\right]_{x=0} \)
I also found a descent approximation to T(a):
\( T(a) \approx \log_{2.55} \log_{2.55} (2.29 + 2.58 a) \)
Andrew Robbins
\(
\begin{tabular}{rl}
\frac{\partial}{\partial a} ({}^{x}{a}) & = \frac{1}{a} \sum_{k=1}^{x} \ln(a)^{k-1} \prod_{j=0}^{k} {}^{{(x-j)}}{a} \\
\frac{\partial}{\partial x} ({}^{x}{a}) & = T(a) \ln(a)^x \prod_{k=1}^{x} {}^{k}{a}
\end{tabular}
\)
where
\( T(a) = \left[\frac{\partial}{\partial x} ({}^{x}{a})\right]_{x=0} \)
I also found a descent approximation to T(a):
\( T(a) \approx \log_{2.55} \log_{2.55} (2.29 + 2.58 a) \)
Andrew Robbins
