Chapter 4

More Derivatives

Chapter Overview

Connections

The letter S is pieced together from curves joined at the red dots. Each piece of the outline is defined by parametric equations x = f (t) and y = g(t) where f and g are cubic polynomials and the parameter t runs from 0 to 1. In a font design program such as Fontlab™ you can drag the control points (blue dots) to change the shape of each piece of the outline. (We show control points for just two pieces of the outline.) As you drag the control points, you are actually changing the coefficients of the polynomials f and g. Using the techniques of Example 6 in Section 4.1 you can show that as long as a joint (or node) lies on the line joining its adjacent control points, the two curves that meet at the node will have the same slope there, making a smooth transition. When you send the letter to an output device like a printer, it uses the parametric equations to render the outline at the highest possible resolution.

Chapter 4 Overview

Now that you have completed a chapter filled with derivative rules and their proofs (most of which we hope you were able to understand rather than simply memorize), you are about to embark on another derivative chapter, this one devoted to a single rule: the Chain Rule. This rule certainly deserves its own chapter, but we also want you to become sufficiently familiar with the rich variety of applications that are accessible once you know how to apply it. You might think of Chapter 3 as the chapter that showed you how to make the bricks; now Chapter 4 will show you how to use those bricks to build houses.

Section 4.1: Chain Rule

Lesson Objectives

Apply the chain rule to find the derivative of a composite function.
Use the chain rule to determine the slopes of curves defined parametrically.

Section 4.2: Implicit Differentiation

Lesson Objectives

Determine derivatives using implicit differentiation.
Use the power rule to find the derivative of a function raised to a rational power of x.

Section 4.3: Derivatives of Inverse Trigonometric Functions

Lesson Objectives

Determine derivatives of inverse functions using the chain rule.
Determine derivatives of inverse trigonometric function.

Section 4.4: Derivations of Exponential and Logarithmic Functions

Lesson Objectives

Calculate derivatives of exponential functions with a base of e.
Calculate derivatives of exponential functions with a base other than e.
Calculate derivatives of natural logarithmic functions.
Calculate derivatives of logarithmic functions with a base other than e.

Unit 4 Test

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