Unit 06 · Lesson 06 · Week 17 · Chain Rule, Implicit & Inverse Functions
Second Derivatives of Implicit Relations
Implicit differentiation treats y as a function of x even when the relation is not solved for y. Every derivative of a y-expression carries a dy/dx factor. This lesson emphasizes second derivatives of implicit relations through symbolic, graphical, numerical, and contextual evidence.
Learning intentions
- Define and use the central calculus language in second derivatives of implicit relations.
- 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
Implicit differentiation treats y as a function of x even when the relation is not solved for y. Every derivative of a y-expression carries a dy/dx factor.
\frac{d}{dx}y^n=ny^{n-1}\frac{dy}{dx}
Worked reasoning
Misconception check: A y-term is not constant during implicit differentiation.
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
Second Derivatives of Implicit Relations
Teacher Answer Key · Analyze, calculate, graph, or justify each result. Show reasoning, intervals, units, and technology evidence when used.
Name
Date
For x²+y²=25, find dy/dx.
Answer: −x/y
Reasoning: Differentiate to 2x+2yy'=0 and solve.
Reasoning: Differentiate to 2x+2yy'=0 and solve.
Find the slope of x²+y²=25 at (3,4).
Answer: −3/4
Reasoning: Substitute the point into −x/y.
Reasoning: Substitute the point into −x/y.
Differentiate xy=6 implicitly.
Answer: y+xy'=0
Reasoning: The product rule is required.
Reasoning: The product rule is required.
Solve the previous result for y'.
Answer: −y/x
Reasoning: Move y and divide by x.
Reasoning: Move y and divide by x.
Why does d(y²)/dx include y'?
Answer: Because y depends on x.
Reasoning: The chain rule applies to y(x).
Reasoning: The chain rule applies to y(x).
What should be done before substituting a point?
Answer: Derive and solve the general slope formula.
Reasoning: Early substitution can erase variable dependence.
Reasoning: Early substitution can erase variable dependence.