differentiating vector with different operation on each elements
Different operations on different elements don't prevent differentiation in any way.
Lets, say we call your above Loss function:
$$\mathcal L=L_1(\mathbf w) + L_2(\mathbf w)$$,
where $\mathbf w$ represents the weights of your model, $L_1$ and $L_2$ are the two loss functions you have defined using the different outputs of your model. The key point is that mathematically, I don't care which output you used to find $L_1$ or $L_2$, just the fact they both somehow depend on the weights of the model.
Thus, if our aim is to differentiate, we simply get (assuming everything in our model is differentiable):
$$\frac{\partial\mathcal L}{\partial\mathbf w} = \frac{\partial L_1}{\partial \mathbf w}+\frac{\partial L_2}{\partial \mathbf w}$$