Unit 05 · Lesson 02 · Week 13 · Differentiation Rules & Higher Derivatives
Product Rule
Differentiation rules compress repeated limit arguments into reliable structures. Each rule should be chosen from the algebraic form of the function and verified through units, graph behavior, or a numerical derivative. This lesson emphasizes product rule through symbolic, graphical, numerical, and contextual evidence.
Learning intentions
- Define and use the central calculus language in product rule.
- Move among symbolic, graphical, numerical, and contextual representations.
- Show complete reasoning with correct notation, domain conditions, units, and intervals.
- Verify a conclusion using a second method or representation.
Essential question
How do derivative rules make complex rates of change efficient, interpretable, and verifiable?
Visual model
Differentiation rules compress repeated limit arguments into reliable structures. Each rule should be chosen from the algebraic form of the function and verified through units, graph behavior, or a numerical derivative.
\frac{d}{dx}x^n=nx^{n-1}(fg)'=f'g+fg'\left(\frac fg\right)'=\frac{f'g-fg'}{g^2}
Worked reasoning
Misconception check: The derivative of a product is not the product of the derivatives.
Three learning pathways
Foundation
Reconstruct definitions, annotate a worked example, and solve representative problems with guided prompts.
Core
Complete mixed representations, justify every major step, and verify exact and approximate results.
Extended
Generalize a result, analyze an error or edge case, and connect the lesson to modelling or proof-informed reasoning.
Communication and modelling
State what is known, identify the theorem or structure used, show symbolic work, interpret the result in context, and explain how the conclusion was checked. Calculator or CAD evidence must include settings, units, and a written interpretation.
Khaemenes Academy · Open-Age Calculus I
Product Rule
Student Worksheet · Analyze, calculate, graph, or justify each result. Show reasoning, intervals, units, and technology evidence when used.
Name
Date
Differentiate f(x)=5x⁴−3x²+7.
Differentiate (x²+1)(x³−2).
Differentiate (x²+1)/x.
Differentiate 5.
Differentiate x^(−3).
How can a symbolic derivative be checked?
Lesson record
Mark this lesson complete only after the worksheet, correction, and reflection are finished.