Calculus I · 2A · review
Unit 2A Review: Derivative Meaning and Computation
Review unit 2a review: derivative meaning and computation with mixed practice and links back to the exact lessons behind each skill.
Section overview
Review, practice, exams, and referenceWhat this section is building
Review unit 2a review: derivative meaning and computation with mixed practice and links back to the exact lessons behind each skill.
A dependable solution separates interpretation, rule selection, calculation, and verification.
Classify first, calculate second, and use answer reveals to diagnose the first incorrect decision.
Reading the solution before making a complete attempt or treating every miss as mere algebra.
Unit 2A Review and Exam Preparation
A Complete Rule-Selection Guide
A derivative problem becomes shorter when you identify its structure before writing algebra. Read the original expression from the outside inward.
Derivative decision tree
• Can the expression be simplified into a sum of powers without changing its domain? If yes, simplify first. • Is the outermost operation a sum or difference? Differentiate term by term. • Is it a product of changing functions? Use the product rule unless a short expansion is clearly easier. • Is it a quotient? Simplify if possible; otherwise use the quotient rule. • Is one function inside another? Use the chain rule for every nonconstant layer. • Are and mixed in one equation? Differentiate implicitly. • Is the function an inverse or inverse trigonometric function? Use the corresponding inverse derivative rule and its domain. • Does the variable appear in both base and exponent, or are many factors multiplied? Use logarithmic differentiation. • Is a second or higher derivative requested? Re-read the structure of the derivative you just found before continuing.
| Structure | First response |
|---|---|
| , radicals, reciprocals | rewrite as powers; use power rule |
| products | product rule or strategic expansion |
| quotients | simplify or quotient rule |
| composition | chain rule |
| implicit equation | differentiate both sides; attach to differentiated -terms |
| inverse function | reciprocal slope at corresponding points |
| variable base and exponent | logarithmic differentiation |
| higher derivative | differentiate the current derivative expression |
Explain before calculating
For each function, name the rules in order without taking the derivative:
A correct answer names product plus chain; quotient plus chain; repeated chain layers; and logarithmic differentiation.
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