Differentiation Calculus Chapter 2. The Derivative and the Tangent Line Problem Calculus 2.1

Differentiation

Calculus Chapter 2

The Derivative and the Tangent Line Problem

Calculus 2.1

Calculus chapter 2 3

A tangent line…

• Circles – is perpendicular to the radial line at a point

• Curves• Touches but does not cross?• Touches or intersects at only one point?• Has a slope equal to the slope of the curve

at that point


Slope of secant line

• As point Q approaches point P, the slope of the secant line approaches the slope of the tangent line.

• slope of the tangent line is said to be the limit of the slope of the secant line

sec

f c x f cm

x


Slope of tangent line

• f must be defined on an open interval containing c

• The limit must exist

0

limx

f c x f cm

x


Linear functions

• Have the same slope at any point• Slope of tangent line agrees with slope

of line using “rise over run” definition.


Example

• Find the slope of the graph of the following function at any point.

5 6f x x


Nonlinear functions

• The slope of the tangent line changes


Example

• Find the slope of the graph of the following function at any point.

21f x x


Vertical tangent lines

• If f is continuous on [a, b], and c is in the interval [a, b], each of the following gives a vertical tangent line

0

limx

f c x f c

x

0limx

f a x f a

x

0

limx

f b x f b

x


Definition of the Derivative of a Function

• The limit must exist

0

limx

f x x f xf x

x


Denoting the derivative

x

f x

dy

dxy

df x

dxD y


Example

• Find the derivative of

2 2 3f x x x


Example

• Find the slope of the graph of the following function at the points (1, 1), (4, 2), and (0,0).

f x x


Alternative form of derivative

• Useful for finding the derivative at one point (not in general)

limx c

f x f cf c

x c


Example

• Find the derivative at c = 1

2 2f x x x


Differentiability implies continuity

• If a function is differentiable at c, then it is continuous at c

• The reverse is not true• A continuous function might have

• A sharp point• A vertical tangent line

• See page 111

Basic Differentiation Rules and Rates of Change

Calculus 2.2


Differentiation Rules

• Allow you to find derivatives without the direct use of the limit definition.

• Can be proven using the limit definition.


Constant Rule

• The derivative of a constant function is 0.

• The slope of a horizontal line is 0.

0dc

dx


The Power Rule

1n ndx nx

dx

3

23

f x x

f x x

2

2

f x x

f x x

0

1

f x x

f x x

f x


The Constant Multiple Rule d

cf x cf xdx

3

2

2

3

3 3

9

f x x

f x x

f x x

1

2

1

2

2

3

2

3

2 1

3 2

1

3

f x x

f x x

f x x

f xx


The Sum and Difference Rules

df x g x f x g x

dx


Examples

• Find the derivatives

2 2 3y t t

3 22 3f x x x x

2 23f s s s

s


Derivatives of sine and cosine functions

• Can be proven using the special trig limits

sin cosd

x xdx

cos sind

x xdx


Examples

• Find the derivatives

siny x x

2cos4sin

3

tf t t


Nonexample

• Can’t pull the two out of the sine function

• A composite function• Need the chain rule – section 2.4

sin 2 2cosd

x xdx


Rates of change

• Derivatives can determine the rate of change of one variable with respect to another


Example

• The area of a circle with a radius r is A = pr2. Find the rate of change of the area with respect to r when r = 2 m.


Position function

• Gives the position of an object as a function of time

s t


Average velocity

distancerate

time


Instantaneous velocity

• Or just velocity• Derivative of the position function

v t s t


Speed

• Absolute value of velocity


Position of free-falling object

• s0 is initial height• v0 is initial velocity• g is acceleration due to gravity

• –32 ft/s2

• –9.8 m/s2

20 0

1

2s t gt v t s


Example

• A ball is thrown straight down from the top of a 220-ft building with an initial velocity of –22 ft/s. What is its velocity after 3 seconds? What is its velocity after falling 108 feet?

The Product and Quotient Rules and Higher-Order Derivatives

Calculus 2.3


Product Rule

• Take the first function times the derivative of the second plus the second function times the derivative of the first.

df x g x f x g x g x f x

dx


Product rule with more than two functions

• Take the derivative of each function times the other functions and add all products

df x g x h x

dx

f x g x h x f x g x h x f x g x h x


Examples

• Find the derivatives using the product rule.

2 32 1 1f x x x x

sinf x x x

5 4 2 3f x x x x


The quotient rule

• Take the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator divided by the square of the denominator.


The quotient rule

2

f x g x f x f x g xd

dx g x g x

LOdHI HIdLO

LOLO


Examples

• Find the derivatives using the quotient rule

3

2

3 2

1

x xf x

x

2 1

3

x xf x

x


Avoiding the quotient rule

• See example 6 on page 130


Derivatives of trig functions

• Can be proven using the quotient rule

2tan secd

x xdx

2cot cscd

x xdx

sec sec tand

x x xdx

csc csc cotd

x x xdx


Higher order derivatives

• Second derivative of the function is the derivative of the first derivative

• Third derivative of the function is the derivative of the second derivative

• Etc.• See page 133


Acceleration

• The derivative of velocity.• The second derivative of position.• Example: find the acceleration of an

object in free fall if the position function is

20 0

1

2s t gt v t s

The Chain Rule

Calculus 2.4


The Chain rule

• One of the most powerful differentiation rules

• Used for composite functions

df g x f g x g x

dx


The Chain Rule

• Differentiate the “outside” function, leaving the inside function alone

• Then multiply it by the derivative of the inside function

• Example:

sin 3y x


Examples

• Differentiate

tan 1y x

2 1y x

32 2 1y x x


General Power Rule

1n ndf x n f x f x

dx


Examples

• Differentiate

2

1

3 1s t

t t

322 1f x x

2

1

2g t

t


Examples

• Differentiate

y x x

33 9f x x x

3 5cosy x x

Implicit Differentiation

Calculus 2.5


Explicit functions

• y is an explicit function of x• Solved for y, or easy to solve for y

3 2y x

12xy


Implicit functions

• Can’t be easily solved for y

22 3 4 2x y y xy

2 9y x


Implicit differentiation

• Used to differentiate implicit functions• To differentiate with respect to x

• For each term that involves x alone, proceed as normal

• For each term that involves y, apply the chain rule


Examples

3dx

dx 3d

ydx

3dx y

dx 2d

xydx


Guidelines

1. Differentiate both sides with respect to x

2. Collect all terms with dy/dx on the left and all other terms on the right.

3. Factor out dy/dx.

4. Solve for dy/dx.


Examples

2 2 2x y y x

cos y x y


Representing a Graph by differentiable functions

2 2 4x y

2 1x y


Example

• Find the slope of the tangent line to the graph at the indicated point

2 3 0 , 1,1x y


Finding the second derivative

• Find the second derivative with respect to x in terms of x and y.

2 2 2 3x y x

Related rates

Calculus 2.6


Finding related rates

• Use the chain rule to find the rates of change of two or more related variables that are changing with respect to time.


Example

• Differentiate with respect to t• Insert known values

2 3

Find when 3 if 2

y x x

dy dxx

dt dt


Example

• The radius r of a sphere is increasing at a rate of 2 inches per minute. Find the rate of change of the volume of the sphere when r = 6 inches.


Example

• A conical tank (with vertex down) is 10 feet across the top and 12 feet deep. If water is flowing into the tank at the rate of 10 cubic feet per minute, find the rate of change of the depth of the water the instant it is 8 feet deep.


More examples

• Pages 153 – 157

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Differentiation Calculus Chapter 2. The Derivative and the Tangent Line Problem Calculus 2.1