$\frac{\partial{y}}{\partial{v_h}}$ for backpropagation chain rule

Question

I'm trying to understand a piece of the backpropagation algorithm and I'm rusty on my summation differentiation.

In one slide, the professor puts down $\frac{\partial{y}}{\partial{v_h}}$ for $y = \sum_h v_hz_h + V_0$ as simply $z_h$. Why is this not $\sum z_h$, what happens to the summation term?

Answer 1

You might be getting confused due to the use of $h$ as both the dummy variable of summation and as the index in the variable that you are differentiating with respect to.

The example below should clarify matters.

Consider $$ \sum_{h=1}^n v_h z_h = v_1z_1 + \cdots + v_jz_j + \cdots + v_nz_n $$ where $1 \leq j \leq n$. Taking the derivative of the above with respect to $v_j$ yields $z_j$.