Offline manual backpropagation - Is it correct?

Question

I want to understand the algorithm of backpropagation so I created this example by myself:

I want to backpropagate through the network offline when

Additionally in last layer we consider linear activation function and MSE as loss function.

My solution

I generally understand the idea behind backpropagation, I want just to assure if I understand how it works in offline case. I have question about two derivatives:

First is about backpropagating error with respect to weight 9

And second is about backpropagation MSE with respect to weight 6:

where neth_2 means input in node h2. Additionally:

Can I please ask you to check whether those derivatives make sense? I'm really not sure how backpropagation works in offline case so it would be very handy for me if you could check it!