Product and Quotient Rules of Differentiation
July 03, 2015
2.3 Product and Quotient Rules of Differentiation
GOALS: 1. For functions that are not simple, recognize
those with products and/or quotients.
2. Learn the Product Rule and apply it properly.
3. Learn the Quotient Rule and apply it properly.
Study 2.3 # 1­7, 11, 15, 17, 21, 22, 23, 24, 29, 31, 37, 39, 43, 53, 57, 59
61, 63, 67, 75, 83, 87, 93, 97, 101
Class Notes: Prof. G. Battaly, Westchester Community College, NY
Calculus Home Page
Homework Part A Part B derivative practice
Calc Web Pages
2.3 Product and Quotient Rules of Differentiation
Class Notes: Prof. G. Battaly, Westchester Community College, NY
Calculus Home Page
© G. Battaly 2015
Homework Part A Part B derivative practice
1
Product and Quotient Rules of Differentiation
July 03, 2015
2.3 Product and Quotient Rules of Differentiation
How do find derivative?
Class Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part A Part B Calculus Home Page
derivative practice
2.3 Product and Quotient Rules of Differentiation
How do find derivative?
What is the wrong approach?
2
(3x )(2x)
Class Notes: Prof. G. Battaly, Westchester Community College, NY
Calculus Home Page
Homework Part A Part B derivative practice
c
© G. Battaly 2015
2
Product and Quotient Rules of Differentiation
July 03, 2015
2.3 Product and Quotient Rules of Differentiation
How do find derivative?
Class Notes: Prof. G. Battaly, Westchester Community College, NY
Calculus Home Page
Homework Part A Part B derivative practice
2.3 Product and Quotient Rules of Differentiation
How do find derivative?
Class Notes: Prof. G. Battaly, Westchester Community College, NY
Calculus Home Page
© G. Battaly 2015
Homework Part A Part B derivative practice
3
Product and Quotient Rules of Differentiation
July 03, 2015
2.3 Product and Quotient Rules of Differentiation
How do find derivative?
Class Notes: Prof. G. Battaly, Westchester Community College, NY
Calculus Home Page
Homework Part A Part B derivative practice
2.3 Product and Quotient Rules of Differentiation
How do find derivative?
Class Notes: Prof. G. Battaly, Westchester Community College, NY
Calculus Home Page
© G. Battaly 2015
Homework Part A Part B derivative practice
4
Product and Quotient Rules of Differentiation
July 03, 2015
2.3 Product and Quotient Rules of Differentiation
How do find derivative?
Class Notes: Prof. G. Battaly, Westchester Community College, NY
Calculus Home Page
Homework Part A Part B derivative practice
2.3 Product and Quotient Rules of Differentiation
What about a product we cannot expand?
We need a rule to handle these.
Class Notes: Prof. G. Battaly, Westchester Community College, NY
Calculus Home Page
© G. Battaly 2015
Homework Part A Part B derivative practice
5
Product and Quotient Rules of Differentiation
July 03, 2015
2.3 Product and Quotient Rules of Differentiation
Product Rule
PRODUCT RULE:
If f and g are both differentiable functions,
then d [ f(x)g(x) ] = f(x) g'(x) + g(x) f '(x) dx
Class Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part A Part B Calculus Home Page
2.3 Product and Quotient Rules of Differentiation
derivative practice
Product Rule
How do find derivative?
d [ f(x)g(x) ] = f(x) g'(x) + g(x) f '(x) dx
Class Notes: Prof. G. Battaly, Westchester Community College, NY
Calculus Home Page
© G. Battaly 2015
Homework Part A Part B derivative practice
6
Product and Quotient Rules of Differentiation
July 03, 2015
2.3 Product and Quotient Rules of Differentiation
Product Rule
How do find derivative?
d [ f(x)g(x) ] = f(x) g'(x) + g(x) f '(x) dx
Class Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part A Part B Calculus Home Page
derivative practice
2.3 Product and Quotient Rules of Differentiation
Product Rule
d [ f(x)g(x) ] = f(x) g'(x) + g(x) f '(x) dx
Class Notes: Prof. G. Battaly, Westchester Community College, NY
Calculus Home Page
© G. Battaly 2015
Homework Part A Part B derivative practice
7
Product and Quotient Rules of Differentiation
July 03, 2015
Product Rule
2.3 Product and Quotient Rules of Differentiation
d [ f(x)g(x) ] = f(x) g'(x) + g(x) f '(x) dx
Class Notes: Prof. G. Battaly, Westchester Community College, NY
Calculus Home Page
Homework Part A Part B derivative practice
Product Rule
2.3 Product and Quotient Rules of Differentiation
d [ f(x)g(x) ] = f(x) g'(x) + g(x) f '(x) dx
Class Notes: Prof. G. Battaly, Westchester Community College, NY
Calculus Home Page
© G. Battaly 2015
Homework Part A Part B derivative practice
8
Product and Quotient Rules of Differentiation
July 03, 2015
2.3 Product and Quotient Rules of Differentiation
Product Rule
d [ f(x)g(x) ] = f(x) g'(x) + g(x) f '(x) dx
= ­ 4 s2 + 4 ­ s2 = 4 ­ 5s2 2s(1/2) 2s(1/2) 2s(1/2)
Class Notes: Prof. G. Battaly, Westchester Community College, NY
Calculus Home Page
Homework Part A Part B derivative practice
Product Rule
2.3 Product and Quotient Rules of Differentiation
d [ f(x)g(x) ] = f(x) g'(x) + g(x) f '(x) dx
Class Notes: Prof. G. Battaly, Westchester Community College, NY
Calculus Home Page
© G. Battaly 2015
Homework Part A Part B derivative practice
9
Product and Quotient Rules of Differentiation
July 03, 2015
Product Rule
2.3 Product and Quotient Rules of Differentiation
d [ f(x)g(x) ] = f(x) g'(x) + g(x) f '(x) dx
Class Notes: Prof. G. Battaly, Westchester Community College, NY
Calculus Home Page
Homework Part A Part B Product Rule
2.3 Product and Quotient Rules of Differentiation
d [ f(x)g(x) ] = f(x) g'(x) + g(x) f '(x) dx
Class Notes: Prof. G. Battaly, Westchester Community College, NY
Calculus Home Page
© G. Battaly 2015
Homework Part A Part B derivative practice
10
Product and Quotient Rules of Differentiation
July 03, 2015
2.3 Product and Quotient Rules of Differentiation
Product Rule
d [ f(x)g(x) ] = f(x) g'(x) + g(x) f '(x) dx
Class Notes: Prof. G. Battaly, Westchester Community College, NY
Calculus Home Page
Homework Part A Part B Quotient Rule
2.3 Product and Quotient Rules of Differentiation
QUOTIENT RULE:
If f and g are both differentiable functions,
then the quotient f is differentiable at all x
g
such that g(x) ≠ 0
and
Class Notes: Prof. G. Battaly, Westchester Community College, NY
Calculus Home Page
© G. Battaly 2015
Homework Part A Part B derivative practice
11
Product and Quotient Rules of Differentiation
July 03, 2015
2.3 Product and Quotient Rules of Differentiation
Quotient Rule
Class Notes: Prof. G. Battaly, Westchester Community College, NY
Calculus Home Page
Homework Part A Part B 2.3 Product and Quotient Rules of Differentiation
Quotient Rule
Class Notes: Prof. G. Battaly, Westchester Community College, NY
Calculus Home Page
© G. Battaly 2015
Homework Part A Part B derivative practice
12
Product and Quotient Rules of Differentiation
July 03, 2015
2.3 Product and Quotient Rules of Differentiation
Quotient Rule
12. f(t) = cos t 3
t
Class Notes: Prof. G. Battaly, Westchester Community College, NY
Calculus Home Page
Homework Part A Part B 2.3 Product and Quotient Rules of Differentiation
derivative practice
Quotient Rule
Quotient Rule
12. f(t) = cos t 3
t
Class Notes: Prof. G. Battaly, Westchester Community College, NY
Calculus Home Page
© G. Battaly 2015
Homework Part A Part B 13
Product and Quotient Rules of Differentiation
July 03, 2015
2.3 Product and Quotient Rules of Differentiation
Quotient Rule
Can use:
1. Simplification 2. Product Rule 3. Quotient Rule
Class Notes: Prof. G. Battaly, Westchester Community College, NY
Calculus Home Page
Homework Part A Part B Quotient Rule
derivative practice
Quotient Rule
Class Notes: Prof. G. Battaly, Westchester Community College, NY
Calculus Home Page
© G. Battaly 2015
Homework Part A Part B 14
Product and Quotient Rules of Differentiation
July 03, 2015
Quotient Rule
2.3 Product and Quotient Rules of Differentiation
y = 1 ­ sin x cos x
d(sin x) = cos x
dx
d(cos x) = ­ sin x
dx
Class Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part A Part B Calculus Home Page
2.3 Product and Quotient Rules of Differentiation
derivative practice
Quotient Rule
one way to find the derivative, though not the easiest
,
Class Notes: Prof. G. Battaly, Westchester Community College, NY
Calculus Home Page
© G. Battaly 2015
Homework Part A Part B derivative practice
15
Product and Quotient Rules of Differentiation
July 03, 2015
Quotient Rule
2.3 Product and Quotient Rules of Differentiation
easier to use trig identities
y = 1 ­ sin x cos x cos x
G: y = sec x ­ tan x F: dy/dx
dy = sec x tan x ­ sec2x
dx
Class Notes: Prof. G. Battaly, Westchester Community College, NY
Calculus Home Page
Homework Part A Part B 2.3 Product and Quotient Rules of Differentiation
co‐functions
d(sin x) = cos x
dx
d(cos x) = ­ sin x
dx
d(tan x) = sec2x
dx
d(cot x) = ­ csc2x
dx
d(sec x) = sec x tan x dx
d(csc x) = ­ csc x cot x dx
Class Notes: Prof. G. Battaly, Westchester Community College, NY
Calculus Home Page
© G. Battaly 2015
Homework Part A Part B 16
Product and Quotient Rules of Differentiation
July 03, 2015
derivative practice
2.3 Product and Quotient Rules of Differentiation
Class Notes: Prof. G. Battaly, Westchester Community College, NY
Calculus Home Page
Homework Part A 2.3 Product and Quotient Rules of Differentiation
Part B derivative practice
Class Notes: Prof. G. Battaly, Westchester Community College, NY
Calculus Home Page
© G. Battaly 2015
Homework Part A Part B 17
Product and Quotient Rules of Differentiation
July 03, 2015
2.3 Product and Quotient Rules of Differentiation
derivative practice
Class Notes: Prof. G. Battaly, Westchester Community College, NY
Calculus Home Page
Homework Part A Part B 2.3 Product and Quotient Rules of Differentiation
derivative practice
Class Notes: Prof. G. Battaly, Westchester Community College, NY
Calculus Home Page
© G. Battaly 2015
Homework Part A Part B 18
Product and Quotient Rules of Differentiation
July 03, 2015
Class Notes: Prof. G. Battaly, Westchester Community College, NY
Calculus Home Page
Homework Part A Part B derivative practice
Class Notes: Prof. G. Battaly, Westchester Community College, NY
Calculus Home Page
© G. Battaly 2015
Homework Part A Part B 19
Product and Quotient Rules of Differentiation
July 03, 2015
Class Notes: Prof. G. Battaly, Westchester Community College, NY
Calculus Home Page
Homework Part A Part B 2.3 Product and Quotient Rules of Differentiation
derivative practice
Quotient Rule
Class Notes: Prof. G. Battaly, Westchester Community College, NY
Calculus Home Page
© G. Battaly 2015
Homework Part A Part B 20
Product and Quotient Rules of Differentiation
July 03, 2015
Quotient Rule
Class Notes: Prof. G. Battaly, Westchester Community College, NY
Calculus Home Page
Homework Part A Part B derivative practice
Class Notes: Prof. G. Battaly, Westchester Community College, NY
Calculus Home Page
© G. Battaly 2015
Homework Part A Part B 21
Product and Quotient Rules of Differentiation
July 03, 2015
Class Notes: Prof. G. Battaly, Westchester Community College, NY
Calculus Home Page
© G. Battaly 2015
Homework Part A Part B 22