Divergence in spherical why take derivative first?

Question

When we are going the divergence thereom we take the derivative before the dot product e.g. in spherical cordinates: $$\nabla\bullet \vec A =(\vec r\bullet \frac{\partial \vec A}{\partial \vec r}+\frac{\vec \theta}{r} \bullet \frac{\partial \vec A}{\partial \vec \theta}+\frac{\vec \phi}{r sin(\theta)} \bullet \frac{\partial \vec A}{\partial \vec \theta})$$ Why is this? i.e. why don't we do the dot product first?