What is the partial derivative $\frac{\partial y}{\partial x_1}$ in this neural network?

Question

The answer is supposed to be -6, but I don't know how to get that.

Also, in a NN, is that 2nd hidden layer possible, where the neurons are not dependent on all the neurons of the previous layer?

Answer 1

From calculus rules: $\frac{\partial y}{\partial x_1}=\frac{\partial y}{\partial r_1}\frac{\partial r_1}{\partial x_1}+\frac{\partial y}{\partial r_2}\frac{\partial r_2}{\partial x_1}=3c+d=-6$. In a NN the above 2nd hidden layer where the neurons are not dependent on all the neurons of the previous layer is possible and it's usually just called a non fully-connected layer.