A derivative calculator that prints only the answer solves the wrong problem. In an exam or a
problem set, nobody asks you for the derivative of x³ sin x — they ask you to produce it, and the
marks are in the sequence of rules, not the final line.
That is why this engine records a named rule at every node of the expression tree. The product rule fires, and it is written down. Inside it, the power rule and the trigonometric rule fire, and they are written down too, in the order a person working by hand would reach them.
What a step contains
Each step carries four things: the rule’s name, the general law it stands on, the expression at that node with the inner derivatives still unevaluated, and what that works out to. The middle line is the one most calculators skip, and it is the one that makes the working checkable — it shows the shape of the rule before the algebra hides it.
The cost of the choice
Recording steps means the differentiator cannot take shortcuts that a purely numerical tool would take. Some simplifications that would produce a tidier answer are deliberately not applied until after the working has been captured. The answer is the same; the road to it stays legible.