For the moment, consider the corresponding problem involving integration. Let $s(x)$ be the explicit solution to the following integral.
$ \displaystyle s(x)=\int_a^x f(t) \, dt $
The function $s'(x)$ is equivalent to the derivative of the integral with respect to it's upper limit and may be expressed in integral form.
$ \displaystyle s'(x)=\partial _x\left(\int_a^x f(t) \, dt\right)=f(a)+\int_a^x f'(t) \, dt $
Now let $s(x)$ be the explicit solution to the following summation.
$ \displaystyle s(x)=\sum _{t=a}^x f(t) $
The function $s'(x)$ is equivalent to the derivative of the summation with respect to it's upper limit. What is the derivative of $s(x)$ expressed in summation form?
$ \displaystyle s'(x)=\partial _x\left(\sum _{t=a}^x f(t)\right)=\ ? $