I don't get the following
$$ \frac{\partial z}{\partial x} = \frac{\partial}{\partial x}\left[\frac{1}{n}\sum_i^n x_i^2\right] = \frac{x}{2} $$
Here x is a list of numbers (matrix or vector).
In my opinion it should be something like $$ \frac{\partial z}{\partial x_i} = \frac{2x_i}{n} $$