Unit 06 · Lesson 02 · Week 16 · Chain Rule, Implicit & Inverse Functions
Chain Rule with Polynomial and Rational Compositions
The chain rule differentiates a composition by multiplying the rate of the outer function with the rate of the inner function. Clear layer identification prevents missing factors. This lesson emphasizes chain rule with polynomial and rational compositions through symbolic, graphical, numerical, and contextual evidence.
Learning intentions
- Define and use the central calculus language in chain rule with polynomial and rational compositions.
- 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 nested, implicit, and inverse relationships change the way derivatives are calculated and interpreted?
Visual model
The chain rule differentiates a composition by multiplying the rate of the outer function with the rate of the inner function. Clear layer identification prevents missing factors.
\frac{d}{dx}f(g(x))=f'(g(x))g'(x)
Worked reasoning
Misconception check: Differentiating only the outer function omits the inner rate and produces the wrong scale.
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
Chain Rule with Polynomial and Rational Compositions
Teacher Answer Key · Analyze, calculate, graph, or justify each result. Show reasoning, intervals, units, and technology evidence when used.
Name
Date
Differentiate (x²+1)^4.
Answer: 8x(x²+1)^3
Reasoning: Differentiate the outer power and multiply by 2x.
Reasoning: Differentiate the outer power and multiply by 2x.
Differentiate sin(3x).
Answer: 3cos(3x)
Reasoning: Outer derivative cosine times inner derivative 3.
Reasoning: Outer derivative cosine times inner derivative 3.
Differentiate e^(x²).
Answer: 2xe^(x²)
Reasoning: Outer exponential times derivative of x².
Reasoning: Outer exponential times derivative of x².
Differentiate ln(5x−1).
Answer: 5/(5x−1)
Reasoning: Use g'/g.
Reasoning: Use g'/g.
Differentiate √(1+x³).
Answer: 3x²/[2√(1+x³)]
Reasoning: Write as a one-half power and apply the chain rule.
Reasoning: Write as a one-half power and apply the chain rule.
Identify outer and inner functions in cos(x²+4).
Answer: Outer cos(u); inner u=x²+4.
Reasoning: Layer identification controls the chain rule.
Reasoning: Layer identification controls the chain rule.