Description

In this section, we completely explore the concept of the derivative of a function. We start with the geometric interpretation of the derivative such as the rate of change of a function, tangent to a graph, and instanteous velocity. We explore how to find derivatives using the basic definition and then improving finding derivatives using the power and sum, product, quotient, chain, and extended power rules. We explore finding derivatives of trigonometric functions and higher order derivatives. We continue our investigation using implicit differentiation, finding differentials and linear approximation and using Newton's Method for finding derivatives numerically.

Sections of this topic include:

1. Rate of Change of a Function, Tangent to a Graph, Instantaneous Velocity

   We begin the study of the derivative, by exploring it's geometric and physical meaning. Groups within this section include: finding the slope fo the tangent line to a graph at a point; finding the equation of the tangent line to a graph at a point; finding the average rate of change; determining values of x of horizontal, vertical and non-existant tangent lines.

2. The Derivative

   In this section, we explore finding derivatives by using the definition of a derivative. Groups of this section include: finding the derivative by definition; finding the derivative by definition, then finding the tangent line to a graph; sketching a graph of the derivative from the graph of a function.

3. Rules of Differentiation I - Power and Sum Rules

   We next commence on developing rules to find derivatives quicker with less effort. Our first rules involve derivatives of constants, sums or differences, and of powers. Groups of this section include: finding the derivative using differentiation rules; finding the equation of the tangent line to a graph; finding points on a graph where the tangent line is horizontal; finding the equation to a normal line to a graph at a point; determing intervals where the derivative is positive or negative; solving problems involving the derivative.

4. Rules of Differentiation II - Product and Quotient Rules

   Continuing our exploration of formulas for derivatives, we develope the product and quotient rules for derivatives. Groups within this section include: finding derivatives using the product and quotient rules; finding derivatives without the aid of the quotient rule; finding equations of tangent lines to a graph; finding points on a graph where the tangent line is horizontal; finding the points on a graph where the tangent line has an indicated slope; using information to evaluate derivatves.

5. Derivatives of the Trigonometric Functions, Some Preliminary Limit Results

   In this section, we explore both limits of trigonometric functions as well as their derivatives. Groups within this section include: finding limits of trigonometric functions; finding the derivatives of trigonometric functions; finding points on trigonometric graphs where the tangent line is horizontal; finding an equation of the tangent line to a trigonometric graph; finding the equation of the normal line to a graph at a point; finding trigonometric derivatives by using derivatives.

6. Rules of Differentiation III - Chain Rule

   We continue our exploration of derivatives by investigating how to find derivatives of composite functions using the Chain Rule. Groups within this section include: finding derivatives using the Chain Rule; finding slopes of tangent lines at a value using the Chain Rule; finding the equation of a tangent line to a graph at a point using the Chain Rule.

7. Higher-Order Derivatives.

   We investigate the question, if we can find a derivative of a function, what prevents us from finding derivatives of derivatives? These are called higher-order derivatives (such as the second derivative, and third derivative, etc.) Groups in this section includes: finding the second derivative; finding higher order derivatives; exploring noth derivatives; finding points on a graph where the second derivative is 0; determining intervals where the second derivative is positive or negative.

8. Implicit Differentation

   With our improved skills of derivatives, we next explore how to find derivatives of equations. Groups within this section include: using implicit differentation to find derivatives; using implicit differentiation to find the indicated derivative; using implicit differentiation to find the derivative at a point; using implicit differentiation to find the derivative at a single value; using implicit differentiation to find equations of tangent lines; using implicit differentiation to find the second derivative; and verifying implicit differentation.

9. Rules of Differentation Iv - Extended Power Rules

   In this section, we find by changing expressions with radicals to expressions with positive or negative rational exponents that we can find derivatives on any expression. Groups within this section include: using extended power rules to find derivatives; using extended power rules to find the second derivative; using extended power rules to find eequations of tangent lines; using extended power rules with implicit differentiation.

10. Differentials and Linear Approximation

    In this section, we explore the notion of differentials and their applications and to write linear approximations of functions near a point. Groups within this section include: finding dy; using differentials to approximate numbers.

11. Newton's Method

    In 1669, Sir Isaac Newton published his method for finding roots of polynomials using numerical means and with the derivative. In this section, we explore various applications of Newton's Method. Groups in this section include: using Newton's Method to find approximations to numbers; using Newton's Method to find approximations of roots to equations.