Unit 3 covers the Chain Rule, differentiation techniques that follow from it, and higher order derivatives. (CED – 2019 p. 67 – 77). These topics account for about 9 – 13% of questions on the AB exam and 4 – 7% of the BC questions.
Topics 3.1 – 3.6
Topic 3.1 The Chain Rule. Students learn how to apply the Chain Rule in basic situations.
Topic 3.2 Implicit Differentiation. The Chain Rule is used to find the derivative of implicit relations.
Topic 3.3 Differentiation Inverse Functions. The Chain Rule is used to differentiate inverse functions.
Topic 3.4 Differentiating Inverse Trigonometric Functions. Continuing the previous section, the ideas of the derivative of the inverse are applied to the inverse trigonometric functions.
Topic 3.5 Selecting Procedures for Calculating Derivatives. Students need to be able to choose which differentiation procedure should be used for any function they are given. This is where you can review (spiral) techniques from Unit 2 and practice those from this unit.
Topic 3.6 Calculating Higher Order Derivatives. Second and higher order derivatives are considered. Also, the notations for higher order derivatives are included here.
Topics 3.2, 3.4, and 3.5 will require more than one class period. You may want to do topic 3.6 before 3.5 and use 3.5 to practice all the differentiated techniques learned so far. The suggested number of 40 – 50-minute class periods is about 10 – 11 for AB and 8 – 9 for BC. This includes time for testing etc.
Posts on these topics include:
This series of posts reviews and expands what students know from pre-calculus about inverses. This leads to finding the derivative of exponential functions, ax, and the definition of e, from which comes the definition of the natural logarithm.
The Derivatives of Exponential Functions and the Definition of e and This pair of posts shows how to find the derivative of an exponential function, how and why e is chosen to help this differentiation.
Logarithms Inverses are used to define the natural logarithm function as the inverse of ex. This follow naturally from the work on inverses. However, integration is involved and this is best saved until later. I will mention it then.
Here are links to the full list of posts discussing the ten units in the 2019 Course and Exam Description.
2019 CED – Unit 4 Contextual Applications of the Derivative Consider teaching Unit 5 before Unit 4
2019 – CED Unit 5 Analytical Applications of Differentiation Consider teaching Unit 5 before Unit 4
2019 – CED Unit 7 Differential Equations Consider teaching after Unit 8
2019 – CED Unit 8 Applications of Integration Consider teaching after Unit 6, before Unit 7
Updated to include the series on inverses – July 7, 2020