Jay Fox's Linear approximation for tetration and slog

[jaydfox]

More importantly, we can demonstrate continuity of the first derivative:

As before, we can examine the limit at \( \log_b^{\circ 2}(e) \) from the right:

\(
\begin{eqnarray}
{\Large \lim_{\normalsize z \to \log_b^{\circ 2}(e)^{+}}}\ {\Large D_{\normalsize z}} \left[ \mathrm{slog}_b(z) \right]
& = & {\Large D_{\normalsize z}} \left[ n + \frac{z-\exp_b^{\circ n}(1)}{\log_b(e)-\log_b^{\circ 2}(e)} \right]_{z=\log_b^{\circ 2}(e)} \\
& = & \left. \frac{1}{\log_b(e)-\log_b^{\circ 2}(e)} \right|_{z=\log_b^{\circ 2}(e)} \\
& = & \frac{1}{\log_b(e)-\log_b^{\circ 2}(e)}
\end{eqnarray}
\)

And then from the left:

\(
\begin{eqnarray}
{\Large \lim_{\normalsize z \to \log_b_{\circ 2}(e)^{-}}}\ {\Large D_{\normalsize z}} \left[ \mathrm{slog}_b(z) \right]
& = & {\Large \lim_{\normalsize z \to \log_b^{\circ 2}(e)^{-}}}\ {\Large D_{\normalsize z}} \left[ \mathrm{slog}_b(b^z) - 1\right] \\
& = & {\Large D_{\normalsize z}} \left[ n - 1 + \frac{b^z-\exp_b^{\circ n}(1)}{\log_b(e)-\log_b^{\circ 2}(e)} \right]_{z=\log_b^{\circ 2}(e)} \\
& = & \left. \frac{b^z \ln(b)}{\log_b(e)-\log_b^{\circ 2}(e)} \right|_{z=\log_b^{\circ 2}(e)} \\
& = & \frac{\frac{\log_b(e)}{\log_b(e)}}{\log_b(e)-\log_b^{\circ 2}(e)} \\
& = & \frac{1}{\log_b(e)-\log_b^{\circ 2}(e)}
\end{eqnarray}
\)

As before, limits at the other end of the interval can also be demonstrated.