Nice blog. I'll be provocative/pedantic for no good reason and say that what's described isn't "calculus" per se, because you can't do calculus on discrete objects like a graph. However, you can define the derivative purely algebraically (as a linear operation which satisfies the Leibniz chain/product rule), which is more accurately what is being described.
You’re not doing calculus on a graph- you’re using a graph algorithm to automate the derivative taking process.
Essentially, you transform your function into a “circuit” or just a graph with edge labels according to the relationship between parts of the expression. The circuit has the nice property that there is an algorithm you can run on it, with very simple rules, which gets you the derivative of the function used to create that circuit.
So taking the derivative becomes:
1. Transform function F into circuit C.
2. Run compute_gradiant(c) to get the gradient of F.
If we're being pedantic, then there's also a more general definition of calculus, which is the first definition in Merriam-Webster: "a method of computation or calculation in a special notation (as of logic or symbolic logic)." One example of this is the lambda calculus. Differential and integral calculus are just special cases of this general definition.
