Lesson 2: 3-2 Implicit Differentiation

What is the difference between explicit and implicit functions?
Why do we need implicit differentiation?
How does implicit differentiation differ from regular differentiation?
What is the chain rule, and how does it apply to implicit differentiation?

Implicit Equation
Explicit Equation
Differentiation with Respect to x
Leibniz Notation
Isolating the Derivative
Power Rule for Implicit Terms
Collect and Factor
Vertical Tangent (Denominator = 0)
Horizontal Tangent (Numerator = 0)
Implicit Second Derivative

F-IF.C.7: Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.  
F-IF.C.8: Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.  
F-TF.A.1: Understand radian measure of an angle as the length of the arc on the unit circle intercepted by the angle.
F-TF.A.2: Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.

This lesson teaches students how to differentiate equations where y is implicitly defined as a function of x. Instead of solving for y first (which is often impossible), students differentiate both sides of the equation with respect to x, treating y as a differentiable function of x. The lesson concludes with isolating dy/dx algebraically.
Purpose: To expand differentiation to relation-based graphs (circles, hyperbolas, etc.) and to prepare students for Related Rates (Lesson 3.6), where almost all differentiation is implicit with respect to time (t).
DOK 1: Recall and Reproduction
Requires students to perform basic calculations and apply the chain rule to differentiate implicit functions.
DOK 2: Skills/Concepts
Involves solving problems that require multiple steps and the application of implicit differentiation to more complex equations.
DOK 3: Strategic Thinking
Demands students to analyze complex scenarios, apply multiple concepts, and justify their reasoning.
DOK 4: Extended Thinking
Requires students to investigate complex problems, formulate models, and create innovative solutions.

For Struggling Learners: Use "Color Coding." Have students write every dy/dx in a bright color (like red) so they don't lose track of it during the algebraic "collect and factor" phase.
For Advanced Learners: Challenge them to find the coordinates of all points where the tangent line is vertical for an ellipse. This requires them to realize they must set the denominator of their derivative expression to zero and solve the resulting system of equations.

AP College Board Classroom Assessments