CHAPTER 2
Differentiation: Basic Concepts
The Derivative
Techniques of Differentiation
Product and Quotient Rules; Higher-Order
Derivatives
The Chain Rule
Marginal Analysis and Approximations
Using Increments
Implicit Differentiation and Related Rates
1
SECTION 2.1 The Derivative
2
Example 2.1.1
The graph shown in Figure 2.2 gives the
relationship between the percentage of
unemployment U and the corresponding
percentage of inflation I. Use the graph to
estimate the rate at which I changes with respect
to U when the level of unemployment is 3% and
again when it is 10%.
3
Example 2.1.2
If air resistance is neglected, an object dropped
from a great height will fall s(t)=16t2 feet in t
seconds.]
a. What is the object’s velocity after t=2 seconds?
b. How is the velocity found in part (a) related to the
graph of s(t)
4
Example 2.1.2
5
6
The Derivative
The expression f ( x + h) − f ( x) is called a difference
h
quotient for f(x)
The Derivative of a Function
The derivative of the function f(x) with respect to x
is the function f’(x) given by
f ' ( x) = lim
h→0
f ( x + h) − f ( x )
h
7
Example 2.1.3
Find the derivative of the function f(x)=16x2 .
8
Example 2.1.4
First compute the derivative of f(x)=x3, and then use it to
find the slope of the tangent line to the curve y=x3 at the
point where x=-1. What is the equation of the tangent line
at this point?
9
Example 2.1.5
A manufacturer determines that when x thousand units of a
particular commodity are produced, the profit generated
will be
P(x)=-400x2+6,800x - 12,000
dollars. At what rate is profit changing with respect to the
level of production x when 9,000 units are produced?
10
11
Significance of the Sign of the Derivative f’(x)
If the function f is differentiable at x=c, then
f is increasing at x=c if f’(c)>0
f is decreasing at x=c if f’(c)<0
12
Example 2.1.6
First compute the derivative of f ( x) = x
then use it to:
a. Find the equation of the tangent line to the curve y = x
at the point where x=4.
b. Find the rate at which y = x is changing with respect
to x when x=1.
13
Differentiability and Continuity
Continuity
of a Differentiable Function
If the function f(x) is differentiable at x=c, then it is
also continuous at x=c.
14
SECTION 2.2 Techniques of
Differentiation
The Constant Rule
For any constant c,
d
[c] = 0
dx
The Power Rule
For any real number n,
d n
[ x ] = nx n−1
dx
15
Example 2.2.2
Verify the power rule for the function F(x)=1/x2 =x-2 by
showing that its derivative is F’(x)=-2x-3.
16
The Constant Multiple Rule
If c is a constant and f(x) is differentiable, then so is
cf(x) and
d
d
[cf ( x)] = c [ f ( x)]
dx
dx
The Sum Rule
If f(x) and g(x) are differentiable, then so is
the sum S(x)=f(x)+g(x) and g’(x)+g’(x); that is,
d
d
d
[ f ( x) + g ( x)] = [ f ( x)] + [ g ( x)]
dx
dx
dx
17
Example 2.2.5
Differentiate the polynomial y=5x3+4x2+12x-8.
18
Example 2.2.6
It is estimated that x months from now, the population
of a certain community will be P(x)=x2+20x+18,000.
a. At what rate will the population be changing with
respect to time 15 months from now?
b. By how much will the population actually change
during the 16th month?
19
Example 2.2.7
The gross domestic product (GDP) of a certain country was
N(t)=t2+5t+106 billion dollars t years after 1995.
a. At what rate was the GDP changing with respect
to time in 2005?
b. At what percentage rate was the GDP changing
with respect to time in 2005?
20
21
Example 2.2.9
The position at time t of an object moving along a line is given s(t)=t36t2+9t+5.
a. Find the velocity of the object and discuss its motion
between times t=0 and t=4.
b. Find the total distance traveled by the object between
times t=0 and t=4.
c. Find the acceleration of the object and determine
when the object is accelerating and decelerating
between times t=0 and t=4.
Interval
Sign of v(t)
Description of the Motion
0<t<1
+
Advances form s(0)=5 to s(1)=9
1<t<3
-
Retreats from s(1)=9 to s(3)=5
3<t<4
+
Advance from s(3)=5 to s(4)=9
22
The Motion of a Projectile
Near sea level, g is approximately 32 ft/sec2.
It can be shown that at time t, the height of the object is
given by the formula
−1 2
H(t ) =
gt + V0t + H 0
2
Where H0 and V0 are the initial height and velocity of the
object
23
Example 2.2.10
Suppose a person standing at the top of a building
112 feet high throws a ball vertically upward with
an initial velocity of 96 ft/sec (see Figure 2.13)
24
SECTION 2.3 Product and Quotient
Higher-Order Derivatives
Rules;
The Product Rule
If f(x) and g(x) are differentiable at x, then so is their
product P(x)=f(x)g(x) and
d
d
d
[ f ( x) g ( x)] = f ( x) [ g ( x)] + g ( x) [ f ( x)]
dx
dx
dx
or equivalently,
( fg )' = fg '+ gf '
25
Example 2.3.1
Differentiate the product P(x)=(x-1)(3x-2) by
a. Expanding P(x) and using the polynomial rule.
b. The product rule.
26
Example 2.3.2
For the curve y=(2x+1)(2x2-x-1):
a. Find y’.
b. Find an equation for the tangent line to the curve at the
point where x=1.
c. Find all points on the curve where the tangent line is
horizontal.
27
Example 2.3.3
A manufacturer determines that t months after a new product
is introduced to the market, x(t)=t2+3t hundred units can be
produced and then sold at a price of p(t)=--2t3/2+30 dollars
per unit.
a. Express the revenue R(t) for this product as a function of
time.
b. At what rate is revenue changing with respect to time after
4 months? Is revenue increasing or decreasing at this time?
28
The Quotient Rule
If f(x) and g(x) are differentiable functions, then so is
the quotient Q(x)=f(x)/g(x) and
d
d
g ( x) [ f ( x)] − f ( x) [ g ( x)]
d  f ( x) 
dx
dx
=
if g ( x) ≠ 0


2
dx  g ( x) 
g ( x)
29
Example 2.3.4
x 2 − 5x + 7
Q ( x) =
2x
Differentiate the quotient
a. Dividing through first.
b. Using the quotient rule.
by
30
Example 2.3.6
Differentiate the function
y=
x 4 x +1
2
−
+ +
2
3 5
x
3x
31
The Second Derivative
The Second Derivative
The second derivative of a function is the derivative
of its derivative . If y=f(x), the second derivative is
denoted by
d2y
or f ' ' ( x)
2
dx
32
Example 2.3.7
Find the second derivative of the function f(x)=5x4-3x2-3x+7.
33
Example 2.3.8
Find the second derivative of y=x2(3x+1)
34
Example 2.3.9(1/2)
An efficiency study of the morning shift at a certain factory
indicates that an average worker who arrives on the job at 8:00
A.M. will have produced Q(t)= -t3+6t2+24t units t hours later.
a.Compute the worker’s rate of production at 11:00 A.M.
b.At what rate is the worker’s rate of production changing
with respect to time at 11:00 A.M.?
35
Higher-Order Derivatives
The nth Derivative
For any positive integer n, the nth derivative of a
function is obtained from the function by
differentiating successively n times. If the original
function is y=f(x), the nth derivative is denoted by
dny
n
or
f
( x)
n
dx
36
Example 2.3.11(1/2)
Find The fifth derivative of each of these functions:
1
a. f ( x) = 4 x 3 + 5 x 2 + 6 x − 1 b.
y=
x
37
SECTION 2.4 The Chain Rule
The Chain Rule
If y=f(u) is a differentiable function of u and u=g(x) is
in turn a differentiable function of x, then the
composite function y=f(g(x)) is a differentiable
function of x whose derivative is given by the product
dy dy du
=
dx du dx
38
Example 2.4.1
Find
dy
if y = ( x 2 + 2) 3 − 3( x 2 + 2) 2 + 1
dx
39
Example 2.4.2
u
where u = 3 x 2 − 1
Consider the function y =
u +1
dy
a. Use the chain rule to find dx
b. Find an equation for the tangent line to the graph of y(x) at
the point where x=1
40
Example 2.4.3
The cost of producing x units of a particular commodity is
C ( x) =
1 2
x + 4 x + 53
3
dollars, and the production level t hours into a particular
production run is x(t)=0.2t2+0.03t units. At what rate is cost
changing with respect to time after 4 hours?
41
The General Power Rule
The General Power rule
For an real number n and differentiable function h,
d
n
n −1 d
[h( x)] = n[h( x)]
[h( x)]
dx
dx
42
Example 2.4.5
Differentiate the function
f ( x) = (2 x 4 − x) 3
43
Example 2.4.6
Differentiate the function
f ( x) = x 2 + 3x + 2
44
Example 2.4.7
Differentiate the function
1
f ( x) =
(2 x + 3)5
45
Example 2.4.8(1/2)
Differentiate the function
f ( x) = (3 x + 1) 4 (2 x − 1) 5
and simplify your answer. Then find all values of x=c for which
the tangent line to the graph of f(x) at (c, f(c)) is horizontal
46
Example 2.4.9(1/2)
Find the second derivative of the function
f ( x) =
3x − 2
( x − 1) 2
47
Example 2.4.10(1/2)
An environmental study of a certain suburban community
suggests that the average daily level of carbon monoxide in the
air will be
c( p ) = 0.5 p 2 + 17
parts per million when the population is p thousand. It is
estimated that t years from now, the population of the
community will be p(t)=3.1+0.1t2 thousand. At what rate will
the carbon monoxide level be changing with respect to time 3
years from now?
48
SECTION 2.5 Marginal Analysis and
Approximations Using Increments
Marginal Cost
If C(x) is the total cost of producing x units of a
commodity, then the marginal cost of producing x0
units is the derivative C’(x0), which approximates the
additional cost C(x0+1)-C(x0) incurred when the level
of production is increased by one unit, from x0 to x0+1
49
Example 2.5.1
A manufacturer estimates that when x units of a particular commodity are
produced, the total cost will be
1
C ( x) = x 2 + 3 x + 98
8
dollars, and furthermore, that all x units will be sold when the price is
1
p ( x ) = (75 − x )
3
dollars per unit.
a. Find the marginal cost and the marginal revenue.
b. Use marginal cost to estimate the cost of producing the ninth unit.
c. What is the actual cost of producing the ninth unit?
d. Use marginal revenue to estimate the revenue derived from the sale of the
ninth unit
e. What is the actual revenue derived from the sale of the ninth unit?
50
Example 2.5.2
A manufacturer of digital cameras estimates that when x
hundred cameras are produced, the total profit will be
p(x)= -0.0035x3+0.07x2+25x-200 thousand dollars.
a. Find the marginal profit function.
b. What is the marginal profit when the level of production is
x=10, x=50, and x=80?
c. Interpret these results.
51
Approximation by Increments
f ( x0 + h) − f ( x0 )
f ( x0 + h) − f ( x0 )
f ' ( x0 ) = lim
, thus f' (x 0 ) ≈
h →0
h
h
Approximation by Increments
If f(x) is differentiable at x=x0 and Δx is a small
change in x, then
f(x0+Δx)≈f(x0)+f’(x0) Δx
or, equivalently, if Δf=f(x0+ Δx)-f(x0), then
Δf ≈ f’(x0)Δx
52
Example 2.5.3
Suppose the total cost in dollars of manufacturing q units of
a certain commodity is C(q)= 3q2+5q+10. If the current
level of production is 40 units, estimate how the total cost
will change if 40.5 units are produced.
53
Example 2.5.5
The daily output at a certain factory is Q(L)=900L1/3 units, where L
denotes the size of the labor force measured in worker-hours. Currently,
1,000 worker-hours of labor are used each day. Use calculus to estimate
the number of additional worker-hours of labor that will be needed to
increase daily output by 15 units.
54
Example 2.5.6
The GDP of a certain country was N(t)=t2+5t+200 billion
dollars t years after 1997. Use calculus to estimate the
percentage change in the GDP during the first quarter of 2005.
55
SECTION 2.6 Implicit Differentiation
and Related Rates
Explicit form: the dependent variable y on the left is
given explicitly by an expression on the right involving
the independent variable x
Implicit form: define y implicitly as a function of x
3
x +1
y=
2x − 3
2
3
x y + 2 y = 3x + 2 y
56
Example 2.6.1
Find
dy
if x 2 y + y 2 = x 3 .
dx
57
Example 2.6.2
Find the slope of the tangent line to the circle x2+y2=25 at the
point (3,4). What is the slope at the point (3, -4)?
58
Example 2.6.3
Find all points on the graph of the equation x2-y2=2x+4y where
the tangent line is horizontal. Does the graph have any vertical
tangents?
59
Example 2.6.4(1/3)
Suppose the output at a certain factory is Q=2x3+x2y+y3 units,
where x is the number of hours of skilled labor used and y is the
number of hours of unskilled labor. The current labor force
consists of 30 hours of skilled labor and 20 hours of unskilled
labor. Use calculus to estimate the change in unskilled labor y
that should be made to offset a 1-hour increase in skilled labor
x so that output will be maintained at its current level.
60
Example 2.6.5
The manager of a company determines that when q hundred units of a
particular commodity are produced, the total cost of production is C
thousand dollars, where C2-3q3=4,275. When 1,500 units are being
produced, the level of production is increasing at the rate of 20 units
per week. What is the total cost at this time and at what rate is it
changing?
61
Example 2.6.8
When the price of a certain commodity is p dollars per unit, the
manufacturer willing to supply x thousand units, where
x 2 - 2 x p − p 2 = 31
How fast is the supply changing when the price is $9 per unit
and is increasing at the rate of 20 cents per week?
62