Unit 05 · Lesson 01 · Week 13 · Differentiation Rules & Higher Derivatives
Constant, Power & Sum Rules
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 constant, power & sum rules through symbolic, graphical, numerical, and contextual evidence.
Learning intentions
- Define and use the central calculus language in constant, power & sum rules.
- 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
Constant, Power & Sum Rules
Teacher Answer Key · Analyze, calculate, graph, or justify each result. Show reasoning, intervals, units, and technology evidence when used.
Name
Date
Differentiate f(x)=4x⁴−3x²+7.
Answer: 16x³−6x
Reasoning: Apply the power rule term by term.
Reasoning: Apply the power rule term by term.
Differentiate (x²+1)(x³−2).
Answer: 2x(x³−2)+3x²(x²+1)
Reasoning: Use the product rule without multiplying derivatives.
Reasoning: Use the product rule without multiplying derivatives.
Differentiate (x²+1)/x.
Answer: (x²−1)/x²
Reasoning: The quotient rule gives [2x·x−(x²+1)]/x².
Reasoning: The quotient rule gives [2x·x−(x²+1)]/x².
Differentiate 5.
Answer: 0
Reasoning: A constant has no change.
Reasoning: A constant has no change.
Differentiate x^(−3).
Answer: −3x^(−4)
Reasoning: Multiply by the exponent and subtract one.
Reasoning: Multiply by the exponent and subtract one.
How can a symbolic derivative be checked?
Answer: Compare with a numerical tangent slope at a sample input.
Reasoning: Independent evidence helps catch algebraic errors.
Reasoning: Independent evidence helps catch algebraic errors.