Note

Now test

\frac{1}{| r - r^{'} |} = \frac{1}{R}

so we can rewrite $R$ as $\sqrt{r^{2} - 2 r \cdot r^{'} + r^{' 2}} = r \sqrt{1 - 2 r \cdot r^{'} / r^{2} + r^{' 2} / r^{2}}$ .
Now use the Taylor expansion

(1 + x)^{- 1 / 2} = 1 - \frac{1}{2} x + O (x^{2})

thus we have

\frac{1}{R} = \frac{1}{r} (1 + \frac{r \cdot r^{'}}{r^{2}} - \frac{r^{' 2}}{2 r^{2}} + O ((\frac{r^{'}}{r})^{2}))

(\frac{r \cdot r^{'}}{r^{2}})^{2} = r^{2} * r^{' 2} / (r^{4}) = r^{' 2} / r^{2}

A second-order Taylor series expansion of a scalar-valued function of more than one variable can be written compactly as

T (x) = f (a) + (x - a)^{T} \nabla f (a) + \dots

where the $x, a$ are vectors