3.5 Selecting Derivative Procedures

Match the structure

Real derivatives combine rules. Identify the structure: sum → term by term; product → product rule; quotient → quotient rule; composite → chain rule; implicit → implicit differentiation.

Nested rules

Functions are often nested (a chain rule inside a product, etc.). Work from the outermost structure inward, applying rules in order. Example: d/dx[x·cos(3x)] needs the product rule and the chain rule.

Key terms

Match the rule

Identify sum, product, quotient, composite, or implicit structure.

Product rule

Use when two functions are multiplied.

Quotient rule

Use when one function is divided by another.

Chain rule

Use for composite (nested) functions.

Nested

Work outermost structure first; combine rules.

Try an AP-style question

d/dx (x² e^(3x)) = ?

1. 2x e^(3x) + x² e^(3x)
2. 6x e^(3x)
3. 2x e^(3x) + 3x² e^(3x)
4. 2x e^(3x)