Using index notation, the last result may be expanded by

$$ \begin{eqnarray*} d^{-1}_{kj}(\mathbf{x^{new}}) & = & d^{-1}_{kj}(\mathbf{x^{old}}) - \sum_{l} \sum_{m} d^{-1}_{km}(\mathbf{x^{new}}) \Delta^{T}_{ml} d^{-1}_{lj}(\mathbf{x^{old}})\\ & = & d^{-1}_{kj}(\mathbf{x^{old}}) - \sum_{l} \sum_{m} d^{-1}_{km}(\mathbf{x^{new}}) \Delta_{lm} d^{-1}_{lj}(\mathbf{x^{cur}})\\ & = & d^{-1}_{kj}(\mathbf{x^{old}}) - \sum_{l} \sum_{m} d^{-1}_{km}(\mathbf{x^{new}}) \delta_{im} (\Delta \phi)_{l} d^{-1}_{lj}(\mathbf{x^{old}})\\ & = & d^{-1}_{kj}(\mathbf{x^{old}}) - d^{-1}_{ki}(\mathbf{x^{new}}) \sum_{l=1}^{N}(\Delta \phi)_{l} d^{-1}_{lj}(\mathbf{x^{old}})\\ & = & d^{-1}_{kj}(\mathbf{x^{old}}) - d^{-1}_{ki}(\mathbf{x^{new}}) \sum_{l=1}^{N}[\phi_{l}(\mathbf{r_{i}^{new}}) - \phi_{l}(\mathbf{r_{i}^{old}})] D^{-1}_{lj}(\mathbf{x^{old}}). \end{eqnarray*} $$