Florida B.E.S.T. MA.912.C.2.5
The Standard
Find the derivatives of implicitly defined functions.
Florida B.E.S.T. Standards for Mathematics
Teacher's field guide
What This Standard Means
What Students Need to Do
- Students differentiate both sides of an equation where y is not isolated. They treat y as a function of x, apply chain and product rules, then solve for dy/dx.
What Mastery Looks Like
- Students correctly apply derivative rules to both sides of an equation and include dy/dx when differentiating terms containing y. They isolate dy/dx and evaluate it at a given point.
Common Misconceptions
- Students often treat y as a constant or write the derivative of y² as 2y. They may forget the product rule for xy or stop before isolating dy/dx.
How to Assess It
- Give this exit ticket: For x² + xy + y² = 7, find dy/dx and the slope at (1, 2).
Lesson moves
Ways to Teach It
Plot points on x² + y² = 25, place a ruler tangent at each point, then compare measured slopes with calculated slopes.
Explain why differentiating y² with respect to x gives 2y(dy/dx), not 2y, using y as a function of x.
Match equation cards with derivative cards, then correct any card that omits a product rule or dy/dx factor.
Use an elliptical tunnel equation to find the wall slope at a marked height and decide how a support beam should meet it.
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