I am reading the Feynman lectures and at this point http://www.feynmanlectures.caltech.edu/I_13.html#Ch13-S3 it says as follows:
The time derivate of the potential energy is
$\begin{equation} \dfrac{d}{dt}\sum\limits_{pairs}-\frac{Gm_{i}m_{j}}{r_{ij}} = \sum\limits_{pairs} \left( +\frac{Gm_{i}m_{j}}{r^2_{ij}} \right) \left( \dfrac{dr_{ij}}{dt} \right) \end{equation}$
But
$\begin{equation} r_{ij}=\sqrt{(x_i-x_j)^2+(y_i-y_j)^2+(z_i-z_j)^2}, \end{equation}$
so that
$\begin{equation} \begin{split} \frac{dr_{ij}}{dt} = \frac{1}{2r_{ij}} \biggl[ & 2 \left( x_{i}-x_{j} \right) \left( \dfrac{dx_{i}}{dt} - \dfrac{dx_{j}}{dt} \right) \\ +& 2 \left( y_{i}-y_{j} \right) \left( \dfrac{dy_{i}}{dt} - \dfrac{dy_{j}}{dt} \right) \\ +& 2 \left( z_{i}-z_{j} \right) \left( \dfrac{dz_{i}}{dt} - \dfrac{dz_{j}}{dt} \right) \biggr] \end{split} \end{equation}$
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What happens after "so that"? I don't understand what is being done mathematically?