# Partial derivative with summation

I need to find
$$\frac{\partial E}{\partial y_i}$$ of $${E = \sum_{k} \frac{e^{y_k}}{\sum_{i} e^{y_i}}}$$
For the fraction I can use the quotient rule, but what happen to $${\frac{\partial E}{\partial y_i} e^{y_k}}$$? Does it become 1 because we need to consider it as a constant?

Note You should not use $$i$$ as both a free index and a bound index. That shall often become a source of confusion. Rather, I suggest using $$j$$ for the series in the denominator.

Well, you could use the quotient rule, but why?

\qquad\begin{align}\dfrac{\sum_k \mathrm e^{y_k}}{\sum_{j}\mathrm e^{y_j}}&=1\end{align}

So, by being a constant, its partial derivative shall be $$0$$.

To confirm, notice that the partial derivative of a series with respect to a particular indexed variable, is the derivative of the lone term belonging to that index. (The derivatives of other-indexed terms are $$0$$, as partial derivatives of functions of an independent variable should).

\qquad\begin{align}\dfrac{\partial \sum_k f(y_k)}{\partial y_i} &= \dfrac{\partial f(y_i)}{\partial y_i}\\[1ex]&=f'(y_i)\end{align}

So when applying the quotient rule, the numerator shall vanish.