It is not necessary to apply the quotient rule, since the function is linear in each $x_i, 1\leq i\leq N$. Let's denote the constant
\begin{align*}
\sum_{i=1}^N\frac{1}{\sigma_i^2}=C
\end{align*}
The function can then be written for $1\leq i\leq N$
\begin{align*}
\overline{x}&=\frac{1}{C}\sum_{j=1}^N\frac{x_j}{\sigma_j^2}\\
&=\frac{1}{C\sigma_i^2}x_i+\frac{1}{C}\sum_{{j=1}\atop{j\ne i}}^N\frac{x_j}{\sigma_j^2}
\end{align*}
which is of the form
\begin{align*}
\overline{x}(x_i)=a x_i + b\qquad\qquad a,b \quad\text{const.}
\end{align*}
The derivative $\frac{\partial}{\partial x_i}\overline{x}$ is therefore
\begin{align*}
\frac{\partial}{\partial x_i}\overline{x}&=\frac{\partial }{\partial x_i}\left(
\frac{1}{C\sigma_i^2}x_i+\frac{1}{C}\sum_{{j=1}\atop{j\ne i}}^N\frac{x_j}{\sigma_j^2}\right)\\
&=\frac{1}{C\sigma_i^2}\qquad\qquad\qquad\qquad\qquad\qquad 1\leq i\leq N
\end{align*}
Note: We have to use a different index variable $j$ for summation to avoid conflicts with the variable $x_i$.
Add-on: [2017-01-22] Some remarks to the index notation with respect to OPs comments.
It seems that Eqn 4.18 in the referred book causes some troubles due to the index notation.
\begin{align*}
\frac{\partial}{\partial x_i}\frac{\sum\left(x_i/\sigma_i^2\right)}{\sum\left(1/\sigma_i^2\right)}
=\frac{1/\sigma_i^2}{\sum\left(1/\sigma_i^2\right)}\tag{1}
\end{align*}
At first we take a look at the left hand side of (1). The expression can be equivalently written as
\begin{align*}
\frac{\partial}{\partial x_i}\frac{\sum\left(x_i/\sigma_i^2\right)}{\sum\left(1/\sigma_i^2\right)}
=\frac{\partial}{\partial x_i}\frac{\sum\left(x_j/\sigma_j^2\right)}{\sum\left(1/\sigma_k^2\right)}
=\frac{\partial}{\partial x_i}\frac{\sum_{j=1}^{N}\left(x_j/\sigma_j^2\right)}{\sum_{k=1}^{N}\left(1/\sigma_k^2\right)}\tag{2}
\end{align*}
When looking at $x_i$ in
\begin{align*}
\sum\left(x_i/\sigma_i^2\right)\tag{3}
\end{align*}
the index $i$ is a so-called bound variable. This means
- the scope of $i$ in (3) belongs to the $\Sigma$ symbol and is limited by the parenthesis
$$\sum\color{blue}{\left(\right.}x_i/\sigma_i^2\color{blue}{\left.\right)}$$
any usage of $i$ somewhere else means another, different symbol
the symbol $i$ in the derivative operator $\frac{\partial}{\partial x_i}$ has nothing to do with the symbol $i$ in (3)
the symbol $i$ in the sum $\sum\left(1/\sigma_i^2\right)$ has nothing to do with $i$ in (3).
Hint: Since the indices $i$ in the two sums and in the derivative operator have nothing in common, it is usually more convenient to use distinct variable names, e.g. $i,j,k$ as we can see in (2).
The expression $\frac{\partial}{\partial x_i}$ stands for the derivative of any $x_i$ arbitrarily, fixed chosen from $1\leq i \leq N$.
Albeit the notation the authors use is mathematically correct, it should be avoided as overloading of the same symbol makes the text harder to read for the less experienced.
