Multivariate partial finite difference product rule

Suppose I have a scalar-valued multivariate function which can be expressed as a product of simpler functions, and I want to compute something like the finite difference "gradient."

I am looking for a good reference on the mechanics of finite differences in this context, and some basic theorems about how to navigate this.

Is the partial difference gradient as well-behaved as the ordinary gradient?

What is the product rule for a large product where finite differences are concerned?

These are fairly basic questions, but I'm having trouble finding these things online. Is there a good textbook (or web page) for this material?

• In general, finite differences don't preserve the (continuous) product rule. This has given rise to so called 'splitting techniques', whereby gradients of products are split into an affine combination of the various terms arising from the different ways in which the (continuous) product rule can be applied. The coefficients of the splitting are determined by some property of the discrete problem (e.g. stability requirements). Burger's equation is a typical example where splittings are used in this way. – ekkilop Jun 2 '17 at 12:26
• What about the product rule given here? johndcook.com/blog/2009/02/01/finite-differences – Mike Battaglia Jun 2 '17 at 13:00
• I realize that I might be answering another question than the one you are asking. Do I understand you properly in that you want to perform a finite difference analog of the continuous product rule, but not necessarily approximate the continuous product rule using finite differences? (my first comment relates to the latter) – ekkilop Jun 3 '17 at 10:58