0
$\begingroup$

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?

$\endgroup$

1 Answer 1

0
$\begingroup$

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.

$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .