For the first derivative only \( N-1 \) terms survive the ratio because the \( g \)-terms that are not differentiated cancel with their corresponding ones in the denominator. Then,
$$ \frac{1}{\Psi_C}\frac{\partial \Psi_C}{\partial x_k} = \sum_{i=1}^{k-1}\frac{1}{g_{ik}}\frac{\partial g_{ik}}{\partial x_k} + \sum_{i=k+1}^{N}\frac{1}{g_{ki}}\frac{\partial g_{ki}}{\partial x_k}. $$An equivalent equation is obtained for the exponential form after replacing \( g_{ij} \) by \( \exp(f_{ij}) \), yielding:
$$ \frac{1}{\Psi_C}\frac{\partial \Psi_C}{\partial x_k} = \sum_{i=1}^{k-1}\frac{\partial g_{ik}}{\partial x_k} + \sum_{i=k+1}^{N}\frac{\partial g_{ki}}{\partial x_k}, $$with both expressions scaling as \( \mathcal{O}(N) \).