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) .