Issue with mathematica derivative with sum

Question

I need to obtain set of derivatives for the function f and so I use the command:

f = z (Sum[x[i], i])^2 + y Sum[x[i], i]+c;
D[f,{{x[i],y}}];

However, the answer I get is {0,0}. The derivatives individually give

D[f, #] & /@ {x[i], y}
= {i y + 2 i z Sum[x[i],i], Sum[x[i],i] }

which makes sense to me.

However, for a more complicated application I need to use a command of the form:

D[function, {vars1_list}, {vars2_list}]

where vars1 and vars2 are list of variables with respect to which I wish to find the derivative. In addition to the above when I try to use Table for the same f with

l1 = {x[i], y}
l2 = {z,c}

Table[D[f, l1[[1]], l2[[j]]], {i, 1, 2}, {j, 1, 2}]

gives me all zeroes again.

Any suggestions ?

Answer 1

Mathematica can take derivatives of finite sums as long as it is a definite sum, i.e., a sum with limits. I will take the liberty of modifying your expression to use a definite sum:

expr = z Sum[x[i], {i, n}]^2 + y Sum[x[i], {i, n}] + c;
expr //TeXForm

$c+y \sum _i^n x(i)+z \left(\sum _i^n x(i)\right){}^2$

Clearly the symbol i here is just a dummy variable, so when taking a derivative it would be better to use a different symbol, say j. Also, Mathematica does not know what the relationship between j and n is, and it also doesn't know that j is an integer. If we add these constraints and differentiate:

res = Simplify[
    D[expr, {{x[j], y}}],
    j \[Element] Integers && 1 <= j <= n
];
res //TeXForm

$\left\{2 z \sum _i^n x(i)+y,\sum _i^n x(i)\right\}$

we get a nice result.