AP® Calculus AB · Unit 3 · Topic 3.5
Selecting Procedures for Calculating Derivatives
Selecting procedures means scanning an expression from the outside in, naming every rule it needs, then applying them in that order instead of stopping after the first one you spot.
Most functions you differentiate are not simple. They are built in layers: a product of several pieces, a quotient, and inside one of those pieces another function tucked inside another. Each layer calls for its own rule. A single term like x2 might need only the power rule, but x2 e3x sin(x) needs the product rule applied twice, and (x2+1)4/(2x-5) needs the quotient rule wrapped around a chain rule. The number of rules you stack matches the number of layers in the expression, nothing more.
To scan an expression, ask what the last operation would be if you were plugging in a number. Is the whole thing a product of factors? Reach for the product rule first, and differentiate each factor in turn, respecting whatever that factor needs internally. Is it one function divided by another? Quotient rule first. Is a factor itself a function stuffed inside another function, like (x2+1)4? That piece needs the chain rule when its turn comes. The outermost operation tells you which rule starts the problem, but it is rarely the only rule you need, and it is not always the last step either: the chain rule's inner derivative often gets computed only after you have already set up the outer structure.
This topic is the payoff of the whole unit. Composite, product, and quotient rules were taught separately so you could see each mechanism cleanly; here you combine them the way real problems demand. The same scanning habit carries forward into implicit differentiation and related rates, where expressions get even more tangled and picking the right rule at each layer is what makes the problem solvable at all.
Key terms
- Scanning
- Reading an expression from the outside in to list every differentiation rule it will require before writing anything down.
- Composition
- A function built by plugging one function into another, such as sin(x2), which needs the chain rule.
- Nested rule
- A rule that must be applied inside a step of another rule, like using the chain rule while finding a quotient rule's numerator derivative.
- Product rule for three factors
- Differentiate each factor once while holding the other two fixed, then add the three resulting terms.
Watch it explained
- How it worksHow do you scan an expression and decide which rule to reach for first?
- Common mistakeWhy isn't the outermost operation always the last one you deal with?
- Worked exampleWalk me through differentiating x2 e3x sin(x) by choosing rules in the right order.
- Worked exampleFind the derivative of (x2 + 1)4/(2x - 5) using the chain and quotient rules.
Check yourself
End of topic.