Analytically express chi-square sum and solve

Question

I have an expression:

chisq[f_, x_, y_, e_, pars__] := Sum[((y[k] - f@@Join[{x[k]},pars])/e[k])^2, {k,1,n}]

which should work for any function that has x as an argument, as well as some arbitrary number of parameters. Say, if I have a and b, chisq[f,x,y,e,{a,b}] will give

which is just an expression of chi-square $\chi^2$ for a set of data $\{x_i, y_i\pm e_i\}_{i=1}^{n}$ against a model described by f[x,a,b].

What I want to do is calculate $\nabla\chi^2 $ to solve $\nabla \chi^2=0$ . If I take derivative against a, D[chisq[f,x,y,e,{a,b}],a], the result is

which is not technically wrong. But I want to arrange this into a format so that it could be solved, by extracting the multiplicative constants out of the summation and splitting the summation into pieces. For example, Simplify[D[chisq[f,x,y,e,{a,b}],a]/2] gives

which has 2 in all of its terms but does not get it out of the sum. What I want it to happen is something like $$ \chi^2 = \sum_{i=1}^{N} \frac{\left(y_i - f(x_i;a,b)\right)^2}{{e_i}^2} \\ \frac{\partial}{\partial a}\chi^2 = 2 \left[\sum_{i=1}^{N} \frac{f(x_i;a,b) \partial_a f(x_i;a,b) - y_i \partial_a f(x_i;a,b)}{{e_i}^2}\right] \\ = 2 \sum_{i=1}^{N} \frac{f(x_i;a,b) \partial_a f(x_i;a,b)}{{e_i}^2} - 2 \sum_{i=1}^{N} \frac{y_i \partial_a f(x_i;a,b)}{{e_i}^2} $$

For example, it should work for linear fit $f(x;a,b) = a+bx$,

$$ \chi^2 = \sum_{i=1}^{N} \frac{\left(y_i - a - b x_i\right)^2}{{e_i}^2} \\ \frac{\partial}{\partial a}\chi^2 = 2 \left[\sum_{i=1}^{N} \frac{a+b x_i}{{e_i}^2} - \sum_{i=1}^{N} \frac{y_i}{{e_i}^2}\right] = 2 \left[a \sum_{i=1}^{N} \frac{1}{{e_i}^2} + b \sum_{i=1}^{N} \frac{x_i}{{e_i}^2} - \sum_{i=1}^{N} \frac{y_i}{{e_i}^2}\right] \\ \frac{\partial}{\partial b}\chi^2 = 2 \left[\sum_{i=1}^{N} \frac{\left(a+b x_i\right) x_i}{{e_i}^2} - \sum_{i=1}^{N} \frac{y_i x_i}{{e_i}^2}\right] = 2 \left[a \sum_{i=1}^{N} \frac{ x_i}{{e_i}^2} + b \sum_{i=1}^{N} \frac{{x_i}^2}{{e_i}^2} - \sum_{i=1}^{N} \frac{y_i x_i}{{e_i}^2}\right] $$

and so on. How can I make the sum D[chisq[f,x,y,e,{a,b}],a] to be expanded like this?

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Edit: I've narrowed down the problem, but not out of the woods yet.

I think I'm halfway there. Thanks to a post https://mathematica.stackexchange.com/a/199464/29247 and with a lot of experimentation, I somehow got hold of some set of rules, which could do what I want when successively applied several times.

First, the one from aforementioned post

Sum[Times[Longest[u___], x_] , {y_, z___}] :> 
    Times[u] Sum[Times[x], {y, z}] /;(FreeQ[{u}, y])

can extract multiplicative factor.

Second,

Sum[Plus[ Longest[u___], x_] , {y_, z___}] :> Plus[Sum[u, {y, z}], Sum[x, {y, z}]]

will split the summation.

Third,

Times[Plus[Longest[u___], v_],x_] :> Times[Plus[u], x] + Times[v, x]

is used for expanding the fraction, since when it's multiplied with common factor, it is not recognized by two rules above.

So, for instance,

Sum[(a[k] + q b[k])/(c[k]), {k,1,n}] /.
    Times[Plus[Longest[u___], v_],x_] :> Times[Plus[u], x] + Times[v, x] /.
    Sum[Plus[ Longest[u___], x_] , {y_, z___}] :> Plus[Sum[u, {y, z}], Sum[x, {y, z}]] /.
    Sum[Times[Longest[u___], x_] , {y_, z___}] :> Times[u] Sum[Times[x], {y, z}] /;(FreeQ[{u}, y])

I get what I want:

But if I make it a set of rules then apply,

Sum[(a[k] + q b[k])/(c[k]), {k,1,n}] //. {
    Times[Plus[Longest[u___], v_],x_] :> Times[Plus[u], x] + Times[v, x] ,
    Sum[Plus[ Longest[u___], x_] , {y_, z___}] :> Plus[Sum[u, {y, z}], Sum[x, {y, z}]] ,
    Sum[Times[Longest[u___], x_] , {y_, z___}] :> Times[u] Sum[Times[x], {y, z}] /;(FreeQ[{u}, y])
}

I get

I figured that it seems the rule from the aforementioned post

Sum[Times[Longest[u___], x_] , {y_, z___}] :> 
    Times[u] Sum[Times[x], {y, z}] /;(FreeQ[{u}, y])

is causing this problem. If I remove it from the rules and apply separately, it seems to work as intended. But I don't understand this behavior. I mean, I'm novice in terms of Wolfram Language, and I understand almost none of what is happening. So, could someone explain why this isn't working, and what I should do to make it work as a single list of rules?