Chapter 2

THE DERIVATIVE

2.1 Two Problems with One Theme

Tangent Line (Euclid)

A tangent is a line touching a curve at just one point.

- Euclid (323 – 285 BC)

Tangent Line (Archimedes)

A tangent to a curve at P is the line that best approximates the curve near P.

- Archimedes (227 – 212 BC)

Examples

1. Find the slope of the tangent lines to the curve of 𝑦 = 𝑓 𝑥 =

− 𝑥2 + 𝑥 + 2 at the points with 𝑥-coordinates −1,1

2, 2,3.

2. Find the equation of the tangent line to the curve 𝑦 =1

𝑥at 2,

1

2.

Average Velocity and Instantaneous Velocity

An object 𝑃 falls in a vacuum. Experiment shows that if it starts from rest, 𝑃 falls 16𝑡2 feet in 𝑡 seconds.

The average velocity of 𝑃 during the interval [1,2] is 16 2 2−16

2−1= 48.

The average velocity of 𝑃 during the interval [1,1.5] is 16 1.5 2−16

1.5−1= 40.

The average velocity of 𝑃 during the interval [1,1.1] is16 1.1 2−16

1.1−1= 33.6.

The average velocity of 𝑃 during the interval [1,1.01] is16 1.01 2−16

1.01−1= 32.16.

Example

An object, initially at rest, falls due to gravity.

a. Find its instantaneous velocity at 𝑡 = 3.8 seconds.

b. How long will it take the object to reach an instantaneous velocity of 112 feet per second?

2.2 The Derivative

The Derivative

If the limit exists, we say 𝑓 is differentiable at 𝑥. Finding a derivative is called differentiationand the part of Calculus associated with derivatives is called Differential Calculus.

Examples.

1. Let 𝑓 𝑥 = 𝑥3 + 7𝑥. Find 𝑓′ 4 .

2. Find 𝐹′ 𝑥 if 𝐹 𝑥 = 𝑥, 𝑥 ≥ 0.

Equivalent Forms for Derivatives

Examples. Each of the following is a derivative, but of what function and at which point?

a. limℎ→0

4+ℎ 2−16

ℎ

b. lim𝑥→3

2

𝑥−2

3

𝑥−3

Differentiability Implies Continuity

Proof?

The converse of Theorem A is not true.

A continuous function is not differentiable at any point where the graph of the function has sharp corner.

Increments and Leibniz Notation

The change in 𝑥 is called an increment of 𝑥 and denoted by ∆𝑥. Corresponding to the increment of 𝑥, we have an increment of 𝑦, ∆𝑦.

The Graph of Derivative

The derivative 𝑓′(𝑥) gives the slope of the tangent line of the graph 𝑦 =𝑓(𝑥) at the value of 𝑥.

Example.

Given the following graph of 𝑦 = 𝑓(𝑥), sketch the graph of 𝑦 = 𝑓′(𝑥)

2.3 Rules for Finding Derivatives

Derivative as an Operator

Three notations for derivative:

𝑓′(𝑥) or 𝐷𝑥𝑓(𝑥) or 𝑑𝑦

𝑑𝑥

The Constant and Power Rules

𝐷𝑥 is a Linear Operator

Example. Find the derivative of 4𝑥6 − 2𝑥4 + 6𝑥3 − 𝑥2 + 𝑥 − 101.

Product and Quotient Rules

Is the derivative of a product, the product of the derivatives?

Examples

1. Find 𝐷𝑥 3𝑥2 − 5 6𝑥4 + 2𝑥

2. Find 𝐷𝑥𝑦 if 𝑦 =2

𝑥4+1+

3

𝑥

3. Show that 𝐷𝑥 𝑥−𝑛 = −𝑛𝑥−𝑛−1

2.4 Derivatives of Trigonometric Function

The Derivatives of sin 𝑥 and cos 𝑥

Examples.

1. Find 𝐷𝑥 𝑥2 sin 𝑥 .

2. Find the equation of the tangent line to the graph of 𝑦 = 3 sin 𝑥 at the point 𝜋, 0 .

The Derivatives of Other Trigonometric Functions

Examples.

1. Find 𝐷𝑥 𝑥𝑛 tan 𝑥 .

2. Find all points in the graph 𝑦 = sin2 𝑥 where the tangent line is horizontal.

2.5 The Chain Rule

Derivative for a composite function

Derivative for a product of functions

Derivative for a quotient of functions

Derivative for a composite of function

Examples

1. If 𝑦 = 2𝑥2 − 4𝑥 + 160, find 𝐷𝑥𝑦.

2. Find 𝐷𝑡𝑡3−2𝑡+1

𝑡4+3

13

.

3. Find 𝐹′(𝑦), where 𝐹 𝑦 = 𝑦 sin 𝑦2 .

4. Find 𝐷𝑥 sin cos 𝑥2 .

2.6 Higher Order Derivatives

Notations for derivatives

Example.

If 𝑦 = sin 2𝑥, find 𝑑12𝑦

𝑑𝑥12

Implicit Differentiation

1. Find 𝑑𝑦/𝑑𝑥 if 4𝑥2𝑦 − 3𝑦 = 𝑥3 − 1.

2. If 𝑠2𝑡 + 𝑡3 = 1, find 𝑑𝑠/𝑑𝑡 and 𝑑𝑡/𝑑𝑠.

3. Sketch the graph of the circle 𝑥2 + 4𝑥 + 𝑦2 + 3 = 0 and then find equations of the two tangent lines that pass through the origin.

Related Rates

1. Each edge of a variable cube is increasing at a rate of 3 inches per second. How fast is the volume of the cube increasing when an edge is 12 inches long?

2. Water is pouring into a conical tank at the rate of 8 cubic feet per minute. If the height of the tank is 12 feet and the radius of its circular opening is 6 feet, how fast is the water level rising when the water is 4 feet deep?

3. An airplane flying north at 640 miles per hour passes over a certain townat noon. A second airplane going east at 600 miles per hour is directly over the same town 15 minutes later. If the airplanes are flying at the same altitude, how fast will they be separating at 1:15 PM?

2.9 Differential and Approximation

∆𝑦 and 𝑑𝑦

If ∆𝑥 small then

∆𝑦 the actual change in 𝑦

𝑑𝑦 an approximation to ∆𝑦

Differentials

Derivative vs Differential

Approximations

Examples.

1. Suppose you need a good approximations to 4.6 and 8.2, but your calculator has died. What might you do?

2. Use differentials to approximate the increase in the area of a soap bubble when its radius increases from 3 inches to 3.025 inches.

Estimating Errors

1. The side of a cube is measured as 11.4 centimeters with a possible error of ±0.05 centimeter. Evaluate the volume of the cube and give an estimate for the possible error in this value.

absolute error vs relative error

2. Poiseuille’s Law for blood flow says that the volume flowing through an artery is proportional to the fourth power of the radius, that is, 𝑣 = 𝑘𝑅4. By how much must the radius be increased in order to increase the blood flow by 50%?

Linear Approximation

𝐿 𝑥 = 𝑓 𝑎 + 𝑓′(𝑎)(𝑥 − 𝑎)

Find and plot the linear approximation to 𝑓 𝑥 = 1 + sin 2𝑥.