calculus and analysis - Can I define a function for vectors of arbitrary dimension?

Is it possible to do analytic calculations with Mathematica? For example, I want to compute:

$$\partial \frac{\sum_{j=1}^n G_{j} \prod_{k=1}^{j-1} (1 - G_{k})}{\partial G_l}=-\prod_{k\neq l} (G_k-1)$$ Is it possible to define such analytic functions for an arbitrary parameter $n$?

Right now, I do it by hand by defining a function like this

f[a_, b_, c_, d_] = a + b (1 - a) + c (1 - a) (1 - b) + d (1 - a) (1 - b) (1 - c)

but I want to do it generally for an $n$-vector, not a 4-vector