Consider the following partial derivative: partial (x*f(x)*g(x)) / partial x

When using the finite difference method (FD) for partial differential equations is it more accurate to apply the partial derivative product rule to the term in parenthesis then use the FD method for each derivative term in the result f(x)g(x) + xgdf/dx + xfdg/dx or to label it say A = x*f(x)g(x) then use the FD method on partial A / partial x in which case we would use say (A[i+1,j] – A[i-1,j])/2dx? Is there a way to prove mathematically say via a Taylor series that one is more accurate than the other?

I wanted to try to mathematically prove one is more accurate than the other perhaps via a Taylor series but was not where to begin.