Unit 06 · Lesson 05 · Week 17 · Chain Rule, Implicit & Inverse Functions
Implicit Differentiation
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 implicit differentiation through symbolic, graphical, numerical, and contextual evidence.
Learning intentions
- Define and use the central calculus language in implicit differentiation.
- 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
Implicit Differentiation
Student Worksheet · 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.
Find the slope of x²+y²=25 at (3,4).
Differentiate xy=6 implicitly.
Solve the previous result for y'.
Why does d(y²)/dx include y'?
What should be done before substituting a point?
Lesson record
Mark this lesson complete only after the worksheet, correction, and reflection are finished.