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If my $$ f(x) = \sum_{i=1}^{n }x_i - \frac{\sum_{t_i\in(R_i)}x_ie^{x_i\beta}}{\sum_{t_i\in(R_i)}e^{x_i\beta}}$$

What is the first derivative of this function with respect to $\beta$ ?

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  • $\begingroup$ Why do you have $\beta_p$ instead of $\beta$? $\endgroup$ – marty cohen Apr 24 at 15:18
  • $\begingroup$ The derivative of the first term is $0$. For the second, use the quotient rule and the chain rules on each summand . (Assuming $\beta = \beta_p$.) $\endgroup$ – Ethan Bolker Apr 24 at 15:20
  • $\begingroup$ @martycohen, thanks marty fixed it, $\endgroup$ – Emily Fassbender Apr 24 at 15:23
  • $\begingroup$ @EthanBolker, thanks Ethan I will try that. $\endgroup$ – Emily Fassbender Apr 24 at 15:24
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    $\begingroup$ The way you've written this, the answer is zero, because summing something over $t_i\in R_i$ when there's no $t_i$ anywhere in the "something" is the same as multiplying by $\lvert R_i\rvert$. After doing the obvious cancellations you no longer have anything dependent on $\beta.$ I suspect the formula you wrote is not the one you want. $\endgroup$ – David K Apr 24 at 15:36
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I will assume that you meant $\beta$ instead of $\beta_p$.

If $f(\beta) = \sum_{i=1}^{n }x_i - \dfrac{\sum_{t_i\in(R_i)}x_ie^{x_i\beta}}{\sum_{t_i\in(R_i)}e^{x_i\beta}} $, then we just apply standard rules of differentiation.

Writing $b$ for $\beta$, $f(b) = \sum_{i=1}^{n }x_i - \dfrac{\sum_{t_i\in(R_i)}x_ie^{x_ib}}{\sum_{t_i\in(R_i)}e^{x_ib}} = \sum_{i=1}^{n }x_i - \dfrac{g(b)}{h(b)} $, where $g(b) =\sum_{t_i\in(R_i)}x_ie^{x_ib} $ and $h(b) =\sum_{t_i\in(R_i)}e^{x_ib} $.

Then $f'(b) =-\dfrac{g'(b)h(b)-g(b)h'(b)}{h^2(b)} $.

The answer is then given by $g'(b) =\sum_{t_i\in(R_i)}x_i^2e^{x_ib} $ and $h'(b) =\sum_{t_i\in(R_i)}x_ie^{x_ib} $.

Nothing complicated.

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  • $\begingroup$ ,Marty is the numerator of $f'(b)$ = 0 ? $\endgroup$ – Emily Fassbender Apr 24 at 15:45
  • $\begingroup$ No. It's a double sum because each of g. h. g', and h' are sums. $\endgroup$ – marty cohen Apr 24 at 15:59

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