differentiating vector with different operation on each elements I have some idea how backpropagation would work for loss function like:
loss=summation(predicted-true)^2
Where predicted and true are vector of same length and same operation throughout the elements.
Now in object localization problem my neural network's output vector's 0th element would denote probability of particular object being in image and rest 4 would tell about bounding box. Now my loss function would be roughly something like this:
loss=CategoricalCrossentropy(pred[0],true[0])+MSE(pred[1:4]-true[1:4])
This loss function worked fine in tensorflow and my NN localized the object as I expected.
My problem is that I am not able to understand  how mathematically differentiation would work when different operations are applied on different 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):
$$\frac{\partial\mathcal L}{\partial\mathbf w} = \frac{\partial L_1}{\partial \mathbf w}+\frac{\partial L_2}{\partial \mathbf w}$$