

\subsection{\bf Derivative Formalism for $\chi^2$---III}

\begin{slide}
\begin{itemize}
\item Now if $t_i$ was evaluated with true theoretical parameters
$\mu_i = 0$ and $\omega_i$ is some number.

\item So $\sigma^2$ is $O(N)$ and $S_1$ is $O(\sqrt{N})$ as desired.

\item Meanwhile, $S_2$ is $O(N)$ as it is sum of $N$ positive numbers.

\item Whence---near solution
\[
\frac{\partial^2 \chi^2}{\partial\alpha^2} = S_2 = 2 \sum^N_{i=1}
\left( \frac{\partial t_i}{\partial\alpha} \right)^2 / \sigma^2_i
\]

\item As claimed: second derivatives of $\chi^2$ only require
evaluation of first derivatives of theory (only true when $N$ large).
\end{itemize}
\end{slide}

