# differentiation with summation symbol

I am trying to understand a step in the math given a scientific paper. They differentiate an objective function of the form:

$$snr = \frac{\sum_{i=1}^n x_it_i}{\sum_{i=1}^n x_id_i}$$

To maximize this function they partially differentiate this function with respect to $x_i$.

$$\frac{\partial{snr} }{\partial{x_i}} = \frac{\dfrac{t_i}{d_i}-\frac{\sum_{j=1}^n x_jt_j}{\sum_{j=1}^n x_jd_j}}{\frac1{d_i}\sum_{j=1}^n x_jt_j}$$

any clues as to how to go that step will help im pretty lost.....

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## 2 Answers

Here's a hint. Suppose we're differentiating with respect to $x_1$. We know that

$snr = \frac{t_1x_1 + s}{d_1x_1 + e}$

where $s = \sum_{j>2} t_jx_j, e =\sum_{j>2} d_jx_j$. Now we can differentiate with respect to $x_1$ by the quotient rule, pretending that the other terms are constants:

$$\frac{\partial snr} {\partial x_1} = \frac{ (t_1)(d_1x_1+e) - (d_1)(t_1x_1+s)}{(d_1x_1+e)^2}$$

The other $x_i$'s are handled very similarly.

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I think there's a mistake in the denominator of the differntiated formula in the science paper. Please check and correct me if I am wrong.

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If you think there is a mistake why don't you give why it is wrong. – Ram Mar 19 '13 at 3:42