02 Derivatives

In this chapter we begin the study of differential calculus, which is concerned with how one quantity changes in relationship to another quantity. The central concept of differential calculus is the derivative, which is an extension of the velocities and slopes of tangents that we studied in Chapter 2. After learning how to calculate derivatives, we use them to solve problems involving rates of change and the approximation of functions.

You will be assessed on the following standards.

2.2   Can sketch a graph of a derivative when given the graph of a function, and vice versa.

2.3   Can compute simple derivatives involving polynomials and fractional powers.

2.5   Can compute derivatives involving trig functions and the chain rule.

Learning Goals for Chapter 2 - Derivatives

Videos and Online Practice Problems

u2.1 Derivatives:

We will extend the concept of tangent lines into the calculus concept of a derivative.

Derivatives presented geometrically, numerically, and algebraically.
Derivatives interpreted as an instantaneous rate of change, and defined as the limit of a difference quotient.
Physical examples of instantaneous rates of change (velocity, reaction rate, marginal cost, and so on) and their units.
Slope of a curve at a point, vertical tangents, no tangents.
Equations for tangent lines to curves and local linear approximations.

Introduction to Differential Calculus

Using Secant Line Slopes to Approximate Tangent Line Slopes

KHW: Using Secant Line Slopes to Approximate Tangent Line Slopes

u2.2 Derivatives as Functions

Armed with the definition of the derivative, we can now discuss what the derivative of a function can tell us about the function, using graphs, values, and equations.

The concept of a differentiable function interpreted graphically, algebraically, and descriptively.
Obtaining the derivative function df/dx by first considering the derivative at a point x, and then treating x as a variable.
How a function can fail to be differentiable.
Sketching the derivative function given a graph of the original function.
Second and higher derivatives

The Derivative: an intuitive introduction

Derivative as the slope of a curve

u2.3 Differentiation Formulas:

We will discover mathematical patterns that help us find derivatives quickly.

The Power, Constant Multiple, Sum and Difference Rules, and how they are developed from the limit definition of the derivative.
Justification of the Product and Quotient Rules.
The computation of derivatives using the above rules.
The definition of the normal line to a curve at a point.

Basic derivative rules

The Power Rule

Differentiating Polynomials

Product Rule

Quotient Rule

Rational Functions

u2.4 Derivatives of Trigonometric Functions

We will learn how to find the derivatives of cyclical or periodic functions. We will analyze the functions and their derivatives graphically and analytically.

Derivatives of basic trig functions, such as sin(x), cos(x), and tan(x).

Differentiating sin(x) and cos(x)

Differentiating tan(x) and cot(x)

u2.5 The Chain Rule

Functions in nature are often complex. We will complete an in-depth exploration using Excel’s numerical and graphical features to derive techniques for finding derivatives of compound functions.

Finding derivatives of compound functions.
A justification of the Chain Rule by interpreting derivatives as rates of change.
The use of the Chain Rule to compute derivatives.

The Chain Rule

Radical Functions

Strategies in differentiating functions

u2.6 Implicit Differentiation

Many functions and models cannot be express in y=f(x) form. Some models may not even be functions (e.g.x2+y2=8). We will learn how to find derivatives of implicit functions.

The concepts of implicit functions and implicit curves.
The technique of implicit differentiation.
The derivatives of the arcsine and arctangent functions.

Implicit Differentiation

KHW: Implicit Differentiation

u2.7 Rates of Change in the Sciences:

We can now apply the concept of derivatives to solving real-world problems, given written descriptions, data, and graphs. We will explain and interpret our answers in the context of our real-world problems.

Instantaneous rate of change as the limit of average rate of change.
Equations involving derivatives, and verbal descriptions translated into equations involving derivatives (and vice versa).
Interpretation of the derivative as a rate of change in varied applied contexts, including velocity, speed, and acceleration.

Equations of normal and tangent linesMotion along a line

u2.8: Related Rates

Using geometry and trigonometry to develop relationships between given and unknown properties of a mathematical system.
Differentiating with special emphasis on the chain rule.
Application of complex problem solving strategies.

We learn how to compute the rate of change of some aspect of a system when some other, often easier to measure, property changes. For example, how does the volume of a balloon change as the diameter changes?

Related Rates

Showing 6 items

HW# and Notes	Recommended Problems
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HW# and Notes	Recommended Problems
2A - Estimating slopes from graphs and finding equations of tangent lines using derivatives are a big aspect of the AP Test.	[u2.1: 1-21 odd, 27, 29, 35, 39, 43, 45] [u2.2: 1, 3, 5, 13, 19, 23, 25, 27, 35, 37, 41, 45]
2B - The rules of differentiation are crucial for the rest of the year. It's vital that you master the rules and practice a lot!	[u2.3: 1-45 odd, 51-92 odd]
2C Trig functions and the Chain Rule - More practice!	[u2.4: 1-23 odd, 29-35 odd, 39-47 odd], [u2.5: 1-45 odd, 75-79, ]
2D - Implicit Derivatives	[u2.6: 1, 7, 9, 13, 17, 23, 25, 27, 35, 37, 49, 59]
2E - Word Problems! Only math teachers give you equations to solve. The real world requires you to convert language into equations and then solve them.	[u2.7: 1-35 odd]
2F - Related Rates (ALWAYS on the AP Test)	[u2.8: 3, 5, 7, 9, 11, 13, 15, 23, 27, 31, 35]