Derivative of a summation

Question

I need to take the derivative of $\sum_{x=1}^{M}\sum_{a=0}^{J-1}\sum_{t=0}^{\infty}u(c(x,t+a,a),l(x,t+a,a))$ for a generic $c(y,s,b)$, or in other words: $$\frac{\partial}{\partial c(y,s,b)} \sum_{x=1}^{M}\sum_{a=0}^{J-1}\sum_{t=0}^{\infty}u(c(x,t+a,a),l(x,t+a,a)).$$ If $y,s,b\in \mathbb{N}, 1\leq y\leq M,0\leq b\leq J-1,0\leq s$, this should give: $$u^{(1,0)}(c(y,s,b),l(y,s,b)). $$ To compute what I need I define:

W = Sum[u[c[x, t + a, a], l[x, t + a, a]], {x, 1, M}, {t, 0, 
   Infinity}, {a, 0, J - 1}]

and then proceed to compute:

D[W, c[y, s, b]]

With this I get $\sum_{x=1}^{M}\sum_{a=0}^{J-1}\sum_{t=0}^{\infty}\delta_{a,b}\delta_{s,a+t}\delta_{x,y}u^{(1,0)}(c(x,t+a,a),l(x,t+a,a))$ while I would like: $$u^{(1,0)}(c(y,s,b),l(y,s,b)) $$ I tried for example to do something like the following to get rid of the Kronecker deltas but without success:

FullSimplify[D[W, c[y, s, b]], 1 < y < M && y \[Element] Integers && 0 < s && s \[Element] Integers  0 < b < J - 1 && b \[Element] Integers]

But it gave me the same result. In other words I'd like Mathematica to get rid of the Kronecker deltas if I specify for example that y,s,b are integer and within the summation indexes.

Answer 1

dwc = D[W, c[y, s, b]]


Assuming[
{
y \[Element] Integers,
    s \[Element] Integers,
    b \[Element] Integers
},
     FullSimplify[dwc]
]