Important Terms and Concepts

  • Parametric function
  • Product rule
  • Even and odd function
  • Quotient rule
  • Differentiability at a point
  • Differentiability on an interval
  • Converse
  • Parametric chain rule
  • Implicit relation
  • Implicit differentiation
  • Inverse of a function
  • Arctangent, Arcsine, etc.
  • Principal branch
  • Implicit differentiation
  • Invertible
  • Contrapositive
  • Parameter
  • Parametric equation
  • Explicit relation
  • Related rate

Learning Outcomes

In this chapter, students will...
  • Differentiate functions whose equations are made by combining other functions.
  • Learn and apply algebraic techniques for differentiating trigonometric functions and their inverses.
  • Perform implicit differentiation and extend the power rule to negative and fractional exponents.
  • Differentiate logarithmic functions.

Chapter 4 Sections

4-1 Combination of Two Functios
Objective: On your own or with a study group, explore the derivatives of functions formed by multiplying, dividing, and composing two functions.
Homework: Problems 1-6 all

4-2 Derivative of a Product of Two Functions
Objective: Given a function that is a product of two other functions, find, in one step, an equation for the derivative function.
Aligned Instructional Resources: Khan Academy Video: "Applying the product rule for derivatives"
Homework - Day 1: Q Problems 1-10, 1-22 all
Homework - Day 2: Problems 23-35 odd

4-3 Derivative of a Quotient of Two Functions
Objective: Given a function whose equation contains a quotient of two other functions, find an equation for the derivative function in one step and simplify the answer.
Homework - Day 1: Q Problems 1-10, Problems 1-15 odd
Homework - Day 2: Problems 17-29 odd, 30, 32

4-4 Derivative of the Other Trigonometric Functions
Objective: Given a function whose equation contains any of the six trigonometric functions, find the equation for the derivative function in one step.
Homework - Day 1: Q Problems 1-10, Problems 1-21 odd
Homework - Day 2: Problems 23-35 odd, 36

4-5 Derivative of Inverse Trigonometric Functions
Objective: Differentiate each of the six inverse trigonometric functions.
Homework - Day 1: Q Problems 1-10, Problems 1-25 odd

4-6 Differentiability and Continuity
Objective: Prove that a differentiable function is continuous, and use this property to prove that certain functions are continuous.
Homework - Day 1: Q Problems 1-10, Problems 1-19 odd, Section 4-5 14, 16
Homework - Day 2: Problems 21-33 odd, Section 4-5 29

4-7 Derivative of a Parametric Function
Objective: Given equations for x and y in terms of t, find dx/dt, dy/dt and dy/dx.
Homework - Day 1: Q Problems 1-10, Problems 1-5 all
Homework - Day 2: Problems 7, 9, 11, 15

4-8 Graphs and Derivatives of Implicit Relations
Objective: Given the equation for an implicit relation, find the derivative of y with respect x, and show by graph that the answer is reasonable.
Homework - Day 1: Q Problems 1-10, Problems 1-12 all
Homework - Day 2: Problems 13-27 odd

4-9 Related Rates
Objective: Given a situation in which several quantities vary, predict the rate at which one of them is changing when you know the other, related rates.
Homework - Day 1: Q Problems 1-10, Problems 1-3 all
Homework - Day 2: Problems 4-7 all
Homework - Day 3: Problems 8-12 all