Calculus I · Unit 2A · review

Unit 2A Review: Derivative Meaning and Computation

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.

Decision

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 $x$ and $y$ 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.

Calculus reference table
Structure	First response
$x^n$ , 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 $y'$ to differentiated $y$ -terms
inverse function	reciprocal slope at corresponding points
variable base and exponent	logarithmic differentiation
higher derivative	differentiate the current derivative expression

Checkpoint

Explain before calculating

For each function, name the rules in order without taking the derivative:

(x^2+1)^5e^x, \qquad \frac{\ln(x^2+1)}{1+x}, \qquad \sin^3(2x), \qquad (x+2)^{x^2}.

A correct answer names product plus chain; quotient plus chain; repeated chain layers; and logarithmic differentiation.

After the explanation

Use the section idea

Reading lens

Mixed practice tests recognition: the page title no longer tells you which rule to use.

Mental model

A dependable solution separates interpretation, rule selection, calculation, and verification.

Decision

Classify first, calculate second, and use answer reveals to diagnose the first incorrect decision.

Common trap

Reading the solution before making a complete attempt or treating every miss as mere algebra.

Check yourself

Can you explain why your method fits before comparing your final answer?

Source & rights

Original instruction with traceable references.

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