UNIT 2: Differentiation: Definition and Fundamental Properties

Developing Understanding of Derivatives

The concept of the derivative is fundamental for determining instantaneous rates of change. To help students grasp how the definition of a derivative uses limits to refine average rates of change, it is beneficial to create opportunities for them to investigate average rates over progressively smaller intervals. Activities such as graphing calculator explorations can demonstrate how different mathematical operations affect the slopes of tangent lines, aiding students in understanding basic rules and properties of differentiation. As students choose which differentiation rules to apply, they should be encouraged to follow the order of operations. Building differentiation skills will prepare students to model realistic instantaneous rates of change in Unit 4 and analyze graphical representations in Unit 5.

Building Mathematical Practices

Practices: 1.E, 2.B, 4.C
Mathematicians recognize that a solution's accuracy depends on the precision of the procedure used. The difference between a correct and incorrect answer often stems from minor arithmetic or procedural errors. For students, applying mathematical procedures—especially differentiation rules—with care and precision can be challenging. Mistakes such as omitting necessary notation (like parentheses) or misapplying the product rule by differentiating each factor separately and multiplying the results are common. The material in Unit 2 serves as a foundational opportunity for students to practice applying procedures accurately and to develop habits of self-correction before errors occur.

This unit also offers a chance to revisit and reinforce the skill of connecting various mathematical representations. Students will encounter derivatives expressed analytically, numerically, graphically, and verbally. Practice in extracting information about the original function f from a graph of its derivative f' can help prevent common misunderstandings, such as confusing the graph of the derivative with the graph of the original function itself.

Preparing for the AP Exam

Students should practice presenting clear mathematical structures that connect their work to definitions or theorems. For example, when asked to estimate the slope of a tangent line to a curve at a specific point using a table of values—as in the 2013 AP Exam Free-Response Question 3 Part A—they must explicitly present the difference quotient:
•    C' (3.5) approximately = (C(4) - C (3)) / (4 - 3) = (12.8 - 11.2) / 1
Failing to present this structure, even with a correct numerical answer, results in lost credit. Similarly, when evaluating the derivative of f(x) = u(x) * v(x) at x = 3, students should write out the product rule: as in f′(3) = u(3) * v′(3) + v(3) * u′(3)
and substitute the appropriate values. It is also important for students to demonstrate how expressions are evaluated using the calculator and to apply specified rounding procedures, such as rounding or truncating to three decimal places. Developing the habit of storing intermediate results in the calculator can help prevent the accumulation of rounding errors throughout calculations.