Selecting Procedures for Calculating Derivatives AB & BC

Every rule in this unit is now available, so the work shifts from executing a procedure to choosing one. The choice is made by reading an expression's outermost operation: what you would do last if you were evaluating it at a number. A sum differentiates term by term, a product takes the product rule, a quotient takes the quotient rule, and a function applied to an inside expression takes the chain rule. Inner pieces are diagnosed afterward, and one expression may need several rules in a definite order.

Two things go wrong. A rule gets applied to a structure it does not fit, most often the chain rule to a sum or the product rule to a power such as $(x+1)^3$. Or a valid rule gets applied to an expression that should have been rewritten first, so a constant denominator becomes a quotient rule, a radical never becomes a fractional exponent, and a logarithm of a product is attacked directly instead of being expanded into a sum. Both are reading failures rather than computational ones.