Product and Quotient Rules
Chapter 2
Differentiation:
Basic Concepts
1
The Concepts
The Derivative
Product and Quotient Rules
Higher-Order Derivatives
The Chain Rule
Marginal Analysis
Implicit Differentiation
2
2.1 The Derivative
3
Why We Learn Differentiation?
Calculus is the mathematics of change, and the primary tool
for studying change is a procedure called differentiation.
In this section, we will introduce this procedure and examine
some of its uses, especially in computing rates of change.
Rate of changes, for example velocity, acceleration, the rate of
growth of a population, and many others, are described
mathematically by derivatives.
4
Illustration
If air resistance is neglected, an object dropped from a
great height will fall s(t ) 16t 2 feet in t seconds.
What is the object’s instantaneous velocity after t=2 seconds?
Average rate of change of s(t) over the time period [2,2+h] is
Vave
distance traveled
s ( 2 h ) s ( 2)
elapsed time
( 2 h) 2
16( 2 h) 2 16( 2) 2
64 16h
h
Compute the instantaneous velocity by the limit
Vins lim Vave lim (64 16h) 64
h 0
h 0
That is, after 2 seconds, the object is traveling at the rate of 64 feet
per second.
5
Rates of Change
How to determine instantaneous rate of change or rate
of change of f(x) at x=c ?
Find the average rate of change of f(x) as x varies from
x=c to x=c+h
rate ave
f ( x ) f ( c h) f ( c ) f ( c h) f ( c )
x
h
( c h) c
Compute the instantaneous rate of change of f(x) at x=c by
finding the limiting value of the average rate as h tends to 0
f ( c h) f ( c )
rate ins lim rate ave lim
h 0
h 0
h
6
Rates of Change (Linear function)
A linear function y(x)=mx+b changes at the constant rate m
with respect to the independent variable x. That is the rate of
change of y(x) is given by the slope or steepness of its graph.
Slope rate of change
change in y
change in x
y2 y1
y
x x 2 x1
7
Rates of Change (Non-Linear function)
For the function that is
nonlinear, the rate of change
is not constant but varies
with x.
In particular, the rate of
change at x=c is given by the
steepness of the graph of f(x)
at the point (c,f(c)), which
can be measured by the slope
of the tangent line to the
graph at p.
8
The Slope of Secant Line
The secant line is a line that intersects the curve at the
point x and point x+h
The average rate of change can be interpreted geometrically as
the slope of the secant line from the point (x,f(x)) to the point
(x+h,f(x+h)).
9
The Derivative
A difference quotient for the function f(x) is the
expression
f ( x h) f ( x )
h
The derivative of a function f(x) with respect to x is the
function f’(x) given by
f ( x h) f ( x )
f ( x) lim
h 0
h
The process of computing the derivative is called
differentiation. f(x) is differentiable at x=c if f’(c) exists
10
Example
Find the derivative of the function f ( x) 2 x 2 16 x 35
The difference quotient for f(x) is
f ( x h) f ( x) 2( x h) 2 16( x h) 35 2 x 2 16 x 35
h
h
4 xh 2h 2 16h
4 x 2h 16
h
Thus, the derivative of f(x) is the function
f ( x h) f ( x )
f ( x) lim
lim (4 x 2h 16) 4 x 16
h 0
h 0
h
11
Slope as a Derivative: The slope of the tangent
line to the curve y=f(x) at the point (c,f(c)) is
mtan f (c)
Instantaneous Rate of Change as a Derivative:
The rate of change of f(x) with respect to x
when x=c is given by f’(c)
Remarks: Since the slope of the tangent line at (a,f(a))
is f’(a), the equation of the tangent line is
y f (a) f (a)( x a)
The point-slope form
12
Example
What is the equation of the tangent line to the curve
y x
at the point where x=4 ?
According to the definition of derivative, we have
f ( x h) f ( x )
xh x
f ( x) lim
lim
h 0
h 0
h
h
( x h x )( x h x )
1
1
lim
lim
h 0
h 0
h( x h x )
xh x 2 x
Thus, the slope of the tangent line to the curve at the
point where x=4 is f’(4)=1/4 . So the point-slope
form is y 2 1 ( x 4) y 1 x 1
4
4
13
Example
A manufacturer determines that when x thousand units of a
particular commodity are produced, the profit generated will be
p( x) 400 x 2 6800 x 12000
dollars. At what rate is profit changing with respect to the level of
production x when 9 thousand units are produced?
We find that
p ( x h) p ( x )
p( x) lim
h 0
h
400( x h) 2 6800( x h) 12000 (400 x 2 6800 x 12000)
lim
h 0
h
400h 2 800hx 6800h
lim
800 x 6800
h 0
h
14
Thus, when the level of production is x=9, the profit is changing
at the rate of p(9) 800(9) 6800 400dollars per
thousand units.
Which means that the tangent line to the profit curve y=p(x) is
sloped downward at point Q where x=9. Therefore, the profit
curve must be falling at Q and profit must be decreasing when
9 thousand units are being produced.
15
Significance of the Sign of the Derivative
If the function f is differentiable at x=c, then
f is increasing at x=c if f’(c)>0 and
f is decreasing at x=c if f’(c)0, p’(x)=0 and
p’(x)0 at x2
p’(x)
Differentiation:
Basic Concepts
1
The Concepts
The Derivative
Product and Quotient Rules
Higher-Order Derivatives
The Chain Rule
Marginal Analysis
Implicit Differentiation
2
2.1 The Derivative
3
Why We Learn Differentiation?
Calculus is the mathematics of change, and the primary tool
for studying change is a procedure called differentiation.
In this section, we will introduce this procedure and examine
some of its uses, especially in computing rates of change.
Rate of changes, for example velocity, acceleration, the rate of
growth of a population, and many others, are described
mathematically by derivatives.
4
Illustration
If air resistance is neglected, an object dropped from a
great height will fall s(t ) 16t 2 feet in t seconds.
What is the object’s instantaneous velocity after t=2 seconds?
Average rate of change of s(t) over the time period [2,2+h] is
Vave
distance traveled
s ( 2 h ) s ( 2)
elapsed time
( 2 h) 2
16( 2 h) 2 16( 2) 2
64 16h
h
Compute the instantaneous velocity by the limit
Vins lim Vave lim (64 16h) 64
h 0
h 0
That is, after 2 seconds, the object is traveling at the rate of 64 feet
per second.
5
Rates of Change
How to determine instantaneous rate of change or rate
of change of f(x) at x=c ?
Find the average rate of change of f(x) as x varies from
x=c to x=c+h
rate ave
f ( x ) f ( c h) f ( c ) f ( c h) f ( c )
x
h
( c h) c
Compute the instantaneous rate of change of f(x) at x=c by
finding the limiting value of the average rate as h tends to 0
f ( c h) f ( c )
rate ins lim rate ave lim
h 0
h 0
h
6
Rates of Change (Linear function)
A linear function y(x)=mx+b changes at the constant rate m
with respect to the independent variable x. That is the rate of
change of y(x) is given by the slope or steepness of its graph.
Slope rate of change
change in y
change in x
y2 y1
y
x x 2 x1
7
Rates of Change (Non-Linear function)
For the function that is
nonlinear, the rate of change
is not constant but varies
with x.
In particular, the rate of
change at x=c is given by the
steepness of the graph of f(x)
at the point (c,f(c)), which
can be measured by the slope
of the tangent line to the
graph at p.
8
The Slope of Secant Line
The secant line is a line that intersects the curve at the
point x and point x+h
The average rate of change can be interpreted geometrically as
the slope of the secant line from the point (x,f(x)) to the point
(x+h,f(x+h)).
9
The Derivative
A difference quotient for the function f(x) is the
expression
f ( x h) f ( x )
h
The derivative of a function f(x) with respect to x is the
function f’(x) given by
f ( x h) f ( x )
f ( x) lim
h 0
h
The process of computing the derivative is called
differentiation. f(x) is differentiable at x=c if f’(c) exists
10
Example
Find the derivative of the function f ( x) 2 x 2 16 x 35
The difference quotient for f(x) is
f ( x h) f ( x) 2( x h) 2 16( x h) 35 2 x 2 16 x 35
h
h
4 xh 2h 2 16h
4 x 2h 16
h
Thus, the derivative of f(x) is the function
f ( x h) f ( x )
f ( x) lim
lim (4 x 2h 16) 4 x 16
h 0
h 0
h
11
Slope as a Derivative: The slope of the tangent
line to the curve y=f(x) at the point (c,f(c)) is
mtan f (c)
Instantaneous Rate of Change as a Derivative:
The rate of change of f(x) with respect to x
when x=c is given by f’(c)
Remarks: Since the slope of the tangent line at (a,f(a))
is f’(a), the equation of the tangent line is
y f (a) f (a)( x a)
The point-slope form
12
Example
What is the equation of the tangent line to the curve
y x
at the point where x=4 ?
According to the definition of derivative, we have
f ( x h) f ( x )
xh x
f ( x) lim
lim
h 0
h 0
h
h
( x h x )( x h x )
1
1
lim
lim
h 0
h 0
h( x h x )
xh x 2 x
Thus, the slope of the tangent line to the curve at the
point where x=4 is f’(4)=1/4 . So the point-slope
form is y 2 1 ( x 4) y 1 x 1
4
4
13
Example
A manufacturer determines that when x thousand units of a
particular commodity are produced, the profit generated will be
p( x) 400 x 2 6800 x 12000
dollars. At what rate is profit changing with respect to the level of
production x when 9 thousand units are produced?
We find that
p ( x h) p ( x )
p( x) lim
h 0
h
400( x h) 2 6800( x h) 12000 (400 x 2 6800 x 12000)
lim
h 0
h
400h 2 800hx 6800h
lim
800 x 6800
h 0
h
14
Thus, when the level of production is x=9, the profit is changing
at the rate of p(9) 800(9) 6800 400dollars per
thousand units.
Which means that the tangent line to the profit curve y=p(x) is
sloped downward at point Q where x=9. Therefore, the profit
curve must be falling at Q and profit must be decreasing when
9 thousand units are being produced.
15
Significance of the Sign of the Derivative
If the function f is differentiable at x=c, then
f is increasing at x=c if f’(c)>0 and
f is decreasing at x=c if f’(c)0, p’(x)=0 and
p’(x)0 at x2
p’(x)