Calculators and numerical differentiation

It happens that I quite like scientific calculators, and lately I’ve been adding a few to my collection. Several calculators are CAS (“Computer Algebra System”)-capable, meaning they are powerful enough to do symbolic / algebraic processing, and thus are able to understand and manipulate algebraic expressions containing any mix of unknowns, operators and numbers, and simplify, expand, and solve those in a general form. They normally can also do symbolic differentiation and integration, computing the general derivative/integral “function” from a expression first, before you can evaluate them at given points, if desired.

But I’ve always found quite fascinating that some less advanced calculators still offer the possibility of doing numerical integration and differentiation. They don’t know for instance that the derivative function of 3⋅x² is 6⋅x, but they can still tell you that when x = 2 the derivative is 12.

Before you ask; I got into this rabbit hole because numerical differentiation and integration are kinda standard features nowadays of any decent calculator. It doesn’t need to be a high end or an expensive calc. In fact, I noticed that even my fx-570MS “entry level” scientific calculator I purchased like 10 years ago can do both operations, while three of my “high end” vintage programmable Casio calculators can’t. As they were “programmable” I think Casio kinda expected users to write their own programs for whatever “advanced” feature they needed. So I started wondering what is the method/algorithm used by current scientific calculators to compute these functions, so I could add the same functionality to my these calcs.

Now, I’ll focus on differentiation here because it’s way easier to deconstruct and understand how that works. A post about integration may follow, if this doesn’t turn out to be a massive borefest.

The concept of derivative

So if you go back to the formal definition of a derivative;

You’ll see that it basically attempts to find the “slope” of the function f(x) at a point x by evaluating the expression at x and (x+h) (as you would do with a straight line), and trying to reduce the difference in the X-axis between those two points (h) to a tiny fraction. In fact, it uses the concept of limit to find out where that expression converges when h approaches 0 (It never really gets there though, because the whole expression goes to hell at that point). Read More