Goldstein's generalized forces

Question

In Goldstein's

$$ Q_{j}=\sum_{i} \mathbf{F}_{i} \cdot \frac{\partial \mathbf{r}_{i}}{\partial q_{j}}=-\sum_{i} \nabla_{i} V \cdot \frac{\partial \mathbf{r}_{i}}{\partial q_{j}} $$ which is exactly the same expression for the partial derivative of a function $-V\left(\mathbf{r}_{1}, \mathbf{r}_{2}, \ldots, \mathbf{r}_{N}, t\right)$ with respect to $q_{j}:$ $$ Q_{j} \equiv-\frac{\partial V}{\partial q_{j}} $$

I don't know why $$-\sum_{i} \nabla_{i} V \cdot \frac{\partial \mathbf{r}_{i}}{\partial q_{j}}=-\frac{\partial V}{\partial q_{j}}.$$

Answer 1

Recall that $\nabla_i V\equiv\frac{\partial V}{\partial {\bf r}_i}$, so OP's last equation is just the chain rule.