Chapter 2: Differentiation
Section 2.1 - What is a Derivative?
Section 2.2 - Precise Definition of the Derivative
Section 2.3 - Differentiation Rules
Section 2.4 - The Chain Rule
Section 2.5 - Implicit Differentiation
Section 2.6 - Derivatives of the Exponential and Logarithmic Functions
Section 2.7 - Derivatives of the Trig Functions
Section 2.8 - The Inverse Trig Functions and Their Derivatives
Section 2.9 - The Hyperbolic Functions and Their Derivatives
Section 2.10 - The Inverse Hyperbolic Functions and Their Derivatives
Straight lines are characterized by their constant slope. The derivative comes into being when the question "Do curves have slope?" is asked. Indeed, at each point on a curve the slope of the line tangent to that point can be assigned as the slope of the curve. The device that makes this assignment is the derivative. Indeed, the derivative can be thought of as the slope of the tangent line along a curve. Of course, this slope changes from point to point, so the derivative is a function obtained from the function whose graph is the curve.
Once a fundamental understanding of the derivative is obtained, efficient rules (algorithms) for obtaining derivatives are needed. These are developed in Chapter 2, after the initial question "What is a derivative?" has been answered. Some calculus texts will proceed to applications that require only the rules for differentiating polynomials, and some calculus texts will define the wider spectrum of the so-called elementary functions and their derivatives.
These elementary functions are xa (which includes the square root), the exponential and logarithmic functions, the six trigonometric functions and their inverses, and the six hyperbolic functions and their inverses. Fortunately, Maple knows all these elementary functions and their inverses. Once the basic formal rules for differentiation are known, Maple can be used to obtain derivatives of all the elementary functions. But that is the story of Chapter 2.
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