Using the identity
$$ \frac{\partial}{\partial x_i}g_{ij} = -\frac{\partial}{\partial x_j}g_{ij}, $$we get expressions where all the derivatives acting on the particle are represented by the second index of \( g \):
$$ \frac{1}{\Psi_C}\frac{\partial \Psi_C}{\partial x_k} = \sum_{i=1}^{k-1}\frac{1}{g_{ik}}\frac{\partial g_{ik}}{\partial x_k} -\sum_{i=k+1}^{N}\frac{1}{g_{ki}}\frac{\partial g_{ki}}{\partial x_i}, $$and for the exponential case:
$$ \frac{1}{\Psi_C}\frac{\partial \Psi_C}{\partial x_k} = \sum_{i=1}^{k-1}\frac{\partial g_{ik}}{\partial x_k} -\sum_{i=k+1}^{N}\frac{\partial g_{ki}}{\partial x_i}. $$