Big Idea 2 from the 2016-2017 AP Calculus Curriculum Framework is Derivatives.
Enduring Understanding 2.1: The derivative of a function is defined as the limit of a difference quotient and can be determined using a variety of strategies.
Learning Objective 2.1C: Calculate Derivatives
Essential Knowledge 2.1C2: Specific rules can be used to calculate derivatives for classes of functions, including polynomial, rational, power, exponential, logarithmic, trigonometric, and inverse trigonometric.
Essential Knowledge 2.1C3: Sums, differences, products, and quotients of functions can be differentiated using derivative rules.
Mathematical Practice for AP Calculus (MPAC) 1: Reasoning with definitions and theorems
Students can: develop conjectures based on exploration with technology.
How do you provide students the opportunity to develop conjectures?
After determining the derivative of a few quadratic functions using the definition, we use our TI-Nspire Computer Algebra System (CAS) software to explore derivatives.
We use the power rule to make conjectures about the product rule. (I think that I saw this suggestion in a Mathematics Teacher magazine in the early 90s, but I can’t find the reference now.)
We know what the derivative should be, because we know the derivative of x^5. How could we use f, f ‘, g, and g ‘ to get to what we know is the derivative from the power rule?
Once students made conjectures about the product rule, we formalized the rule.
I asked students to predict the derivative of f(x)=sin(3x). As expected, many thought that it would be f ’(x)=cos(3x). When we looked at the graph of the derivative of f(x), students realized that f ‘(x)=3cos(3x).
We used CAS to explore the chain rule (power and composite) in more detail.
Students practiced “Notice and Note”. Several generalized the chain power rule before I asked.
Once students knew the chain rule, we used the chain rule to derive the quotient rule.
And so the journey providing opportunities for students to make sense of rules instead of just telling them rules continues …