Lesson 23
Overview
Derivative
Rules
The Basics
The Power Rule
The Product
Rule
The Quotient
Rule
The Chain Rule
Exam 2 Review
Approximation
by Increments
Implicit
Differentiation
Related Rates
March 7th, 2014
Lesson 23
Overview
Derivative
Rules
The Basics
The Power Rule
The Product
Rule
The Quotient
Rule
The Chain Rule
Approximation
by Increments
Implicit
Differentiation
Related Rates
Topics for Exam 2
Lesson 23
Overview
Derivative
Rules
The Basics
The Power Rule
The Product
Rule
The Quotient
Rule
The Chain Rule
Approximation
by Increments
Implicit
Differentiation
Related Rates
Topics for Exam 2
Constant, Constant Multiple, Sum, and Power Rule
Lesson 23
Overview
Derivative
Rules
The Basics
The Power Rule
The Product
Rule
The Quotient
Rule
The Chain Rule
Approximation
by Increments
Implicit
Differentiation
Related Rates
Topics for Exam 2
Constant, Constant Multiple, Sum, and Power Rule
Product Rule
Lesson 23
Overview
Derivative
Rules
The Basics
The Power Rule
The Product
Rule
The Quotient
Rule
The Chain Rule
Approximation
by Increments
Implicit
Differentiation
Related Rates
Topics for Exam 2
Constant, Constant Multiple, Sum, and Power Rule
Product Rule
Quotient Rule
Lesson 23
Overview
Derivative
Rules
The Basics
The Power Rule
The Product
Rule
The Quotient
Rule
The Chain Rule
Approximation
by Increments
Implicit
Differentiation
Related Rates
Topics for Exam 2
Constant, Constant Multiple, Sum, and Power Rule
Product Rule
Quotient Rule
Chain Rule
Lesson 23
Overview
Derivative
Rules
The Basics
The Power Rule
The Product
Rule
The Quotient
Rule
The Chain Rule
Approximation
by Increments
Implicit
Differentiation
Related Rates
Topics for Exam 2
Constant, Constant Multiple, Sum, and Power Rule
Product Rule
Quotient Rule
Chain Rule
Higher-Order Derivatives
Lesson 23
Overview
Derivative
Rules
The Basics
The Power Rule
The Product
Rule
The Quotient
Rule
The Chain Rule
Approximation
by Increments
Implicit
Differentiation
Related Rates
Topics for Exam 2
Constant, Constant Multiple, Sum, and Power Rule
Product Rule
Quotient Rule
Chain Rule
Higher-Order Derivatives
Approximation by Increments
Lesson 23
Overview
Derivative
Rules
The Basics
The Power Rule
The Product
Rule
The Quotient
Rule
The Chain Rule
Approximation
by Increments
Implicit
Differentiation
Related Rates
Topics for Exam 2
Constant, Constant Multiple, Sum, and Power Rule
Product Rule
Quotient Rule
Chain Rule
Higher-Order Derivatives
Approximation by Increments
Implicit Differentiation
Lesson 23
Overview
Derivative
Rules
The Basics
The Power Rule
The Product
Rule
The Quotient
Rule
The Chain Rule
Approximation
by Increments
Implicit
Differentiation
Related Rates
Types of Problems
Lesson 23
Overview
Types of Problems
Derivative
Rules
The Basics
The Power Rule
The Product
Rule
The Quotient
Rule
The Chain Rule
Approximation
by Increments
Implicit
Differentiation
Related Rates
Compute the derivative
Lesson 23
Overview
Types of Problems
Derivative
Rules
The Basics
The Power Rule
The Product
Rule
The Quotient
Rule
The Chain Rule
Approximation
by Increments
Implicit
Differentiation
Related Rates
Compute the derivative
Find the equation of the tangent line
Lesson 23
Overview
Types of Problems
Derivative
Rules
The Basics
The Power Rule
The Product
Rule
The Quotient
Rule
The Chain Rule
Approximation
by Increments
Implicit
Differentiation
Related Rates
Compute the derivative
Find the equation of the tangent line
Find where the tangent line is horizontal
Lesson 23
Overview
Types of Problems
Derivative
Rules
The Basics
The Power Rule
The Product
Rule
The Quotient
Rule
The Chain Rule
Approximation
by Increments
Implicit
Differentiation
Related Rates
Compute the derivative
Find the equation of the tangent line
Find where the tangent line is horizontal
Word Problems
Lesson 23
Overview
Types of Problems
Derivative
Rules
The Basics
The Power Rule
The Product
Rule
The Quotient
Rule
The Chain Rule
Compute the derivative
Find the equation of the tangent line
Find where the tangent line is horizontal
Approximation
by Increments
Word Problems
Implicit
Differentiation
Estimating changes
Related Rates
Lesson 23
Overview
Types of Problems
Derivative
Rules
The Basics
The Power Rule
The Product
Rule
The Quotient
Rule
The Chain Rule
Compute the derivative
Find the equation of the tangent line
Find where the tangent line is horizontal
Approximation
by Increments
Word Problems
Implicit
Differentiation
Estimating changes
Related Rates
Related Rates
The Basics
Lesson 23
Overview
Derivative
Rules
The Basics
The Power Rule
The Product
Rule
The Quotient
Rule
The Chain Rule
Approximation
by Increments
Implicit
Differentiation
Related Rates
The Basics
Lesson 23
The Constant Rule
For any constant c,
Overview
Derivative
Rules
The Basics
The Power Rule
The Product
Rule
The Quotient
Rule
The Chain Rule
Approximation
by Increments
Implicit
Differentiation
Related Rates
d
dx [c]
= 0.
The Basics
Lesson 23
The Constant Rule
For any constant c,
d
dx [c]
= 0.
Overview
Derivative
Rules
The Basics
The Power Rule
The Product
Rule
The Quotient
Rule
The Chain Rule
Approximation
by Increments
Implicit
Differentiation
Related Rates
The Constant Multiple Rule
If c is a constant and f is a differentiable function, then
d
[c f (x)] = c f 0 (x).
dx
The Basics
Lesson 23
The Constant Rule
For any constant c,
d
dx [c]
= 0.
Overview
Derivative
Rules
The Basics
The Power Rule
The Product
Rule
The Quotient
Rule
The Chain Rule
The Constant Multiple Rule
If c is a constant and f is a differentiable function, then
d
[c f (x)] = c f 0 (x).
dx
Approximation
by Increments
Implicit
Differentiation
Related Rates
The Sum Rule
If f (x) and g (x) are differentiable, then
d
[f (x) + g (x)] = f 0 (x) + g 0 (x).
dx
The Power Rule
Lesson 23
Overview
Derivative
Rules
The Basics
The Power Rule
The Product
Rule
The Quotient
Rule
The Chain Rule
Approximation
by Increments
Implicit
Differentiation
Related Rates
The Power Rule
Lesson 23
The Power Rule
Overview
Derivative
Rules
The Basics
The Power Rule
The Product
Rule
The Quotient
Rule
The Chain Rule
Approximation
by Increments
Implicit
Differentiation
Related Rates
For any real number n,
d
n
dx [x ]
= nx n−1 .
The Power Rule
Lesson 23
The Power Rule
Overview
For any real number n,
d
n
dx [x ]
= nx n−1 .
Derivative
Rules
The Basics
The Power Rule
The Product
Rule
The Quotient
Rule
The Chain Rule
Approximation
by Increments
Implicit
Differentiation
Related Rates
Example
Find the equation of the line tangent to f (x) = 4x 3 −
at the point where x = 1.
a. y = 18x − 11
b. y = 13x − 6
c. y = 11x − 4
d. y = 7x
e. y = 16x − 9
√2
x
+5
The Power Rule
Lesson 23
The Power Rule
Overview
For any real number n,
d
n
dx [x ]
= nx n−1 .
Derivative
Rules
The Basics
The Power Rule
The Product
Rule
The Quotient
Rule
The Chain Rule
Example
Find the equation of the line tangent to f (x) = 4x 3 −
at the point where x = 1.
Approximation
by Increments
Implicit
Differentiation
Related Rates
b. y = 13x − 6
√2
x
+5
The Product Rule
Lesson 23
Overview
Derivative
Rules
The Basics
The Power Rule
The Product
Rule
The Quotient
Rule
The Chain Rule
Approximation
by Increments
Implicit
Differentiation
Related Rates
The Product Rule
Lesson 23
The Product Rule
If f (x) and g (x) are differentiable at x, then
Overview
Derivative
Rules
The Basics
The Power Rule
The Product
Rule
The Quotient
Rule
The Chain Rule
Approximation
by Increments
Implicit
Differentiation
Related Rates
d
[f (x)g (x)] = f 0 (x)g (x) + f (x)g 0 (x).
dx
The Product Rule
Lesson 23
The Product Rule
If f (x) and g (x) are differentiable at x, then
Overview
d
[f (x)g (x)] = f 0 (x)g (x) + f (x)g 0 (x).
dx
Derivative
Rules
The Basics
The Power Rule
The Product
Rule
The Quotient
Rule
The Chain Rule
Approximation
by Increments
Implicit
Differentiation
Related Rates
Example
Find all points on the graph of f (x) = (x + 3)(x 2 − 6x − 6)
where the tangent line is horizontal.
a.
b.
c.
d.
e.
x
x
x
x
x
= −3, 6
=3
= −2, 6
= −4, 2
= −2, 4
The Product Rule
Lesson 23
The Product Rule
If f (x) and g (x) are differentiable at x, then
Overview
d
[f (x)g (x)] = f 0 (x)g (x) + f (x)g 0 (x).
dx
Derivative
Rules
The Basics
The Power Rule
The Product
Rule
The Quotient
Rule
The Chain Rule
Approximation
by Increments
Implicit
Differentiation
Example
Find all points on the graph of f (x) = (x + 3)(x 2 − 6x − 6)
where the tangent line is horizontal.
Related Rates
e. x = −2, 4
The Quotient Rule
Lesson 23
Overview
Derivative
Rules
The Basics
The Power Rule
The Product
Rule
The Quotient
Rule
The Chain Rule
Approximation
by Increments
Implicit
Differentiation
Related Rates
The Quotient Rule
Lesson 23
Overview
Derivative
Rules
The Basics
The Power Rule
The Product
Rule
The Quotient
Rule
The Chain Rule
Approximation
by Increments
Implicit
Differentiation
Related Rates
The Quotient Rule
If f (x) and g (x) are differentiable at x and g (x) 6= 0, then
d f (x)
g (x)f 0 (x) − f (x)g 0 (x)
=
.
dx g (x)
(g (x))2
The Quotient Rule
Lesson 23
Example
Overview
Compute the derivative of g (x) =
Derivative
Rules
The Basics
The Power Rule
The Product
Rule
The Quotient
Rule
The Chain Rule
Approximation
by Increments
Implicit
Differentiation
a. g 0 (x) =
b. g 0 (x) =
c. g 0 (x) =
Related Rates
d. g 0 (x) =
e. g 0 (x) =
8x 2 − 22x + 5
4x − 1
7 + 2x − 4x 2
(4x − 1)2
4x 2 − 2x − 7
(4x − 1)2
12x 2 − 42x + 17
(4x − 1)2
1
5
2x − 4
x 2 − 5x + 3
.
4x − 1
The Quotient Rule
Lesson 23
Example
Overview
Compute the derivative of g (x) =
Derivative
Rules
The Basics
The Power Rule
The Product
Rule
The Quotient
Rule
The Chain Rule
Approximation
by Increments
Implicit
Differentiation
Related Rates
c. g 0 (x) =
4x 2 − 2x − 7
(4x − 1)2
x 2 − 5x + 3
.
4x − 1
The Chain Rule
Lesson 23
Overview
Derivative
Rules
The Basics
The Power Rule
The Product
Rule
The Quotient
Rule
The Chain Rule
Approximation
by Increments
Implicit
Differentiation
Related Rates
The Chain Rule
Lesson 23
Overview
Derivative
Rules
The Basics
The Power Rule
The Product
Rule
The Quotient
Rule
The Chain Rule
Approximation
by Increments
Implicit
Differentiation
Related Rates
The Chain Rule
If y = f (u) and u = g (x) are differentiable functions, then the
derivative of y = f (g (x)) is given by
dy du
dy
=
dx
du dx
or
d
[f (g (x))] = f 0 (g (x))g 0 (x).
dx
The Chain Rule
Lesson 23
Example
Overview
Compute the second derivative of y = (2x 2 + 7)4
Derivative
Rules
The Basics
The Power Rule
The Product
Rule
The Quotient
Rule
The Chain Rule
a.
b.
Approximation
by Increments
Implicit
Differentiation
c.
Related Rates
d.
e.
d 2y
dx 2
d 2y
dx 2
d 2y
dx 2
d 2y
dx 2
d 2y
dx 2
= 192x 2 (2x 2 ) + 72
= 16x(2x 2 + 7)3
= 4(−4x 2 )3
= 12(2x 2 + 7)2
= 112(2x 2 + 7)2 (2x 2 + 1)
The Chain Rule
Lesson 23
Example
Overview
Compute the second derivative of y = (2x 2 + 7)4
Derivative
Rules
The Basics
The Power Rule
The Product
Rule
The Quotient
Rule
The Chain Rule
Approximation
by Increments
Implicit
Differentiation
Related Rates
e.
d 2y
= 112(2x 2 + 7)2 (2x 2 + 1)
dx 2
Approximation by Increments
Lesson 23
Overview
Derivative
Rules
The Basics
The Power Rule
The Product
Rule
The Quotient
Rule
The Chain Rule
Approximation
by Increments
Implicit
Differentiation
Related Rates
Approximation by Increments
Lesson 23
Overview
Derivative
Rules
The Basics
The Power Rule
The Product
Rule
The Quotient
Rule
The Chain Rule
Approximation
by Increments
Implicit
Differentiation
Related Rates
Approximation by Increments
Given a function y = f (x), we estimate the change in f , ∆f , is
given by
∆f ≈ f 0 (x0 )∆x
where ∆x is the change in x and x0 is the initial value of x.
Approximation by Increments
Lesson 23
Example
Overview
Derivative
Rules
The Basics
The Power Rule
The Product
Rule
The Quotient
Rule
The Chain Rule
Approximation
by Increments
Implicit
Differentiation
Related Rates
An efficiency study of the morning shift at a certain factory
indicates that an average worker arriving on the job at 8:00
A.M. will have assembled f (x) = −2x 3 + 11x 2 + 8x units x
hours later. Approximately how many units will the worker
assemble between 9:00 and 9:10 A.M.?
a. 4
b. − 140
3
c. 2.4
d. 240
e. 40
Approximation by Increments
Lesson 23
Example
Overview
Derivative
Rules
The Basics
The Power Rule
The Product
Rule
The Quotient
Rule
The Chain Rule
Approximation
by Increments
Implicit
Differentiation
Related Rates
An efficiency study of the morning shift at a certain factory
indicates that an average worker arriving on the job at 8:00
A.M. will have assembled f (x) = −2x 3 + 11x 2 + 8x units x
hours later. Approximately how many units will the worker
assemble between 9:00 and 9:10 A.M.?
a. 4
Implicit Differentiation
Lesson 23
Overview
Derivative
Rules
The Basics
The Power Rule
The Product
Rule
The Quotient
Rule
The Chain Rule
Approximation
by Increments
Implicit
Differentiation
Related Rates
Say we are assuming that y is defined implicity as a function of
x. In using implicit differentiation, we must remember to
multiply by dy
dx every time we take the derivative of a term
involving y .
Lesson 23
Example
Compute
dy
dx
by implicit differentiation.
Overview
x 3 − y 2 = 4xy 2
Derivative
Rules
The Basics
The Power Rule
The Product
Rule
The Quotient
Rule
The Chain Rule
Approximation
by Increments
a.
b.
Implicit
Differentiation
Related Rates
c.
d.
e.
dy
dx
dy
dx
dy
dx
dy
dx
dy
dx
3x 2
8xy + 2y
1
=
8xy − 2y
=
= 4y 2 + 2y − 3x 2
=
3x 2 − 4y 2
8xy + 2y
= 4y 2 + 8xy + 2y − 3x 2
Lesson 23
Example
Compute
dy
dx
by implicit differentiation.
Overview
x 3 − y 2 = 4xy 2
Derivative
Rules
The Basics
The Power Rule
The Product
Rule
The Quotient
Rule
The Chain Rule
Approximation
by Increments
Implicit
Differentiation
Related Rates
d.
dy
3x 2 − 4y 2
=
dx
8xy + 2y
Lesson 23
Example
Overview
Derivative
Rules
The Basics
The Power Rule
The Product
Rule
The Quotient
Rule
The Chain Rule
Approximation
by Increments
Implicit
Differentiation
Related Rates
A 6-foot-tall man walks at the rate of 4 ft/sec away from the
base of a street light 12 feet above the ground. At what rate is
the length of his shadow changing when he is 20 feet away
from the base of the light?
a. 4 ft/sec
b. 2 ft/sec
c. 20 ft/sec
d. 8 ft/sec
e. 6 ft/sec
Lesson 23
Example
Overview
Derivative
Rules
The Basics
The Power Rule
The Product
Rule
The Quotient
Rule
The Chain Rule
Approximation
by Increments
Implicit
Differentiation
Related Rates
A 6-foot-tall man walks at the rate of 4 ft/sec away from the
base of a street light 12 feet above the ground. At what rate is
the length of his shadow changing when he is 20 feet away
from the base of the light?
a. 4 ft/sec