Unit 3 – Differentiation

Unit 3 Differentiation: Composite, Implicit, and Inverse Functions.

This unit develops the rules for differentiating more complicated functions using the Chain Rule for composite functions, the rules and relationships of functions and their inverses, and implicitly defined relations.

Composite Functions

Foreshadowing the Chain Rule – When the product rule gets tedious.

Power Rule Implies the Chain Rule – Preparing for the Chain Rule.

The Chain Rule – Derivative Rule IV – Ways to teach the Chain Rule.

Experimenting with CAS – Chain Rule – Helping students discover the chain rule.

Seeing the Chain Rule – A graphical look at the Chain Rule.

Derivative Practice – Numbers – Practice problems using table of numbers to find the values of a derivative.

Derivative Practice – Graphs – Practice problems using table of numbers to find the values of a derivative.

Inverses

 Inverses – The basic idea leads to two concerns.

 Writing Inverses – Concern 1: What if you can’t solve for y?

 The Range of the Inverse – Concern 2: What if the inverse is not a function?

The Calculus of Inverses – Inverse Trigonometric Functions.

Inverses Graphically and Numerically – How the derivative of a function and the derivative of its inverse are related.

Logarithms – Defining the logarithm function as the inverse of the exponential function  

Improper Integrals and Proper Areas – On the range of the inverse tangent.

Arbitrary Ranges – Inverse trigonometric functions and their ranges.

Implicit Functions

Good Question 3 1995 BC 5 – A great graphing calculator questions on families of functions

Implicit Differentiation – When and how to do it.