Concept preview

What you will be able to do

1. 01Read the structure of a differentiation problem and name its TOP-LEVEL operation — a sum, a product, a quotient, a composition, or an implicit relation — before touching any rule.
2. 02Choose the correct primary rule from that top-level structure, then scan each remaining piece for an inner function that demands the chain rule.
3. 03Combine rules cleanly in layered expressions (a product whose factor is a composite, a quotient with a nested angle) without dropping an inner factor or applying the wrong rule.

Every differentiation problem has an outermost operation — the last thing you would do if you were EVALUATING the expression at a number. That outermost operation is your top-level structure, and it dictates your primary rule. Ask, in order: Is the whole thing a sum or difference of separate terms? Then differentiate term by term. Is it one quantity multiplied by another? Use the product rule. Is it one quantity divided by another? Use the quotient rule. Is it one function wrapped inside another — a sine OF something, a square root OF something, a power OF something more complicated than $x$? Use the chain rule. Is $y$ tangled together with $x$ so you cannot isolate it? Differentiate implicitly. Naming this outer layer is the single most important decision you make.