I've been trying to understand de la Vallee Poussin's "Demonstration Simplifiee du Theorem de Dirichlet sur la Progression Arithmetique" and I've got stuck at the following step on pg 18 where Poussin takes the logarithmic derivative of: $\Sigma_{n}\frac{\chi(n)}{n^{s}} = \prod_{q}(1-\frac{\chi(q)}{q^{s}})^{-1}$ in order to obtain: $-D\log\sum_{n}\frac{\chi(n)}{n^{s}} = \Sigma_{q}\frac{\chi(q)\log(q)}{q^{s}-\chi(q)}$
Specifically, when I try to work through the calculation, I don't see where the -1 on the left hand side of the equation comes from. When I tried to take logs and then differentiate the first equation, I ended up with the following:
$D\log\sum_{n}\frac{\chi(n)}{n^{s}} = \Sigma_{q}\frac{\chi(q)\log(q)}{q^{s}-\chi(q)}$
Could someone help me understand where I'm going wrong please?