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Product and Quotient Rules of Differentiation
© G. Battaly 2015 1
July 03, 2015
Calc Web Pages
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
2.3 Product and Quotient Rules of Differentiation
Calculus Home Page
Class Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part A Part B derivative practice
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.
2.3 Product and Quotient Rules of Differentiation
Calculus Home Page
Class Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part A Part B derivative practice
Product and Quotient Rules of Differentiation
© G. Battaly 2015 2
July 03, 2015
How do find derivative?
2.3 Product and Quotient Rules of Differentiation
Calculus Home Page
Class Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part A Part B derivative practice
How do find derivative?
What is the wrong approach?
(3x2)(2x)
2.3 Product and Quotient Rules of Differentiation
Calculus Home Page
Class Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part A Part B derivative practice
c
Product and Quotient Rules of Differentiation
© G. Battaly 2015 3
July 03, 2015
How do find derivative?
2.3 Product and Quotient Rules of Differentiation
Calculus Home Page
Class Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part A Part B derivative practice
How do find derivative?
2.3 Product and Quotient Rules of Differentiation
Calculus Home Page
Class Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part A Part B derivative practice
Product and Quotient Rules of Differentiation
© G. Battaly 2015 4
July 03, 2015
How do find derivative?
2.3 Product and Quotient Rules of Differentiation
Calculus Home Page
Class Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part A Part B derivative practice
How do find derivative?
2.3 Product and Quotient Rules of Differentiation
Calculus Home Page
Class Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part A Part B derivative practice
Product and Quotient Rules of Differentiation
© G. Battaly 2015 5
July 03, 2015
How do find derivative?
2.3 Product and Quotient Rules of Differentiation
Calculus Home Page
Class Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part A Part B derivative practice
What about a product we cannot expand?
We need a rule to handle these.
2.3 Product and Quotient Rules of Differentiation
Calculus Home Page
Class Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part A Part B derivative practice
Product and Quotient Rules of Differentiation
© G. Battaly 2015 6
July 03, 2015
2.3 Product and Quotient Rules of Differentiation
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
Calculus Home Page
Class Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part A Part B derivative practice
Product Rule
How do find derivative?
2.3 Product and Quotient Rules of Differentiation
Calculus Home Page
Class Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part A Part B
d [ f(x)g(x) ] = f(x) g'(x) + g(x) f '(x) dx
derivative practice
Product Rule
Product and Quotient Rules of Differentiation
© G. Battaly 2015 7
July 03, 2015
How do find derivative?
2.3 Product and Quotient Rules of Differentiation
Calculus Home Page
Class Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part A Part B
d [ f(x)g(x) ] = f(x) g'(x) + g(x) f '(x) dx
derivative practice
Product Rule
2.3 Product and Quotient Rules of Differentiation
Calculus Home Page
Class Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part A Part B
d [ f(x)g(x) ] = f(x) g'(x) + g(x) f '(x) dx
derivative practice
Product Rule
Product and Quotient Rules of Differentiation
© G. Battaly 2015 8
July 03, 2015
2.3 Product and Quotient Rules of Differentiation
Calculus Home Page
Class Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part A Part B
d [ f(x)g(x) ] = f(x) g'(x) + g(x) f '(x) dx
derivative practice
Product Rule
2.3 Product and Quotient Rules of Differentiation
Calculus Home Page
Class Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part A Part B
d [ f(x)g(x) ] = f(x) g'(x) + g(x) f '(x) dx
derivative practice
Product Rule
Product and Quotient Rules of Differentiation
© G. Battaly 2015 9
July 03, 2015
2.3 Product and Quotient Rules of Differentiation
= 4 s2 + 4 s2 = 4 5s2 2s(1/2) 2s(1/2) 2s(1/2)
Calculus Home Page
Class Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part A Part B
d [ f(x)g(x) ] = f(x) g'(x) + g(x) f '(x) dx
derivative practice
Product Rule
2.3 Product and Quotient Rules of Differentiation
Calculus Home Page
Class Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part A Part B
d [ f(x)g(x) ] = f(x) g'(x) + g(x) f '(x) dx
derivative practice
Product Rule
Product and Quotient Rules of Differentiation
© G. Battaly 2015 10
July 03, 2015
2.3 Product and Quotient Rules of Differentiation
Calculus Home Page
Class Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part A Part B
d [ f(x)g(x) ] = f(x) g'(x) + g(x) f '(x) dx
Product Rule
2.3 Product and Quotient Rules of Differentiation
Calculus Home Page
Class Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part A Part B
d [ f(x)g(x) ] = f(x) g'(x) + g(x) f '(x) dx
derivative practice
Product Rule
Product and Quotient Rules of Differentiation
© G. Battaly 2015 11
July 03, 2015
Product Rule2.3 Product and Quotient Rules of Differentiation
Calculus Home Page
Class Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part A Part B
d [ f(x)g(x) ] = f(x) g'(x) + g(x) f '(x) dx
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 gsuch that g(x) ≠ 0 and
Calculus Home Page
Class Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part A Part B derivative practice
Quotient Rule
Product and Quotient Rules of Differentiation
© G. Battaly 2015 12
July 03, 2015
2.3 Product and Quotient Rules of Differentiation
Calculus Home Page
Class Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part A Part B
Quotient Rule
2.3 Product and Quotient Rules of Differentiation
Calculus Home Page
Class Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part A Part B derivative practice
Quotient Rule
Product and Quotient Rules of Differentiation
© G. Battaly 2015 13
July 03, 2015
12. f(t) = cos t t3
2.3 Product and Quotient Rules of Differentiation
Calculus Home Page
Class Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part A Part B derivative practice
Quotient Rule
Quotient Rule
12. f(t) = cos t t3
2.3 Product and Quotient Rules of Differentiation
Calculus Home Page
Class Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part A Part B
Quotient Rule
Product and Quotient Rules of Differentiation
© G. Battaly 2015 14
July 03, 2015
2.3 Product and Quotient Rules of Differentiation
Can use:1. Simplification 2. Product Rule 3. Quotient Rule
Calculus Home Page
Class Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part A Part B derivative practice
Quotient Rule
Quotient Rule
Calculus Home Page
Class Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part A Part B
Quotient Rule
Product and Quotient Rules of Differentiation
© G. Battaly 2015 15
July 03, 2015
2.3 Product and Quotient Rules of Differentiation
Calculus Home Page
Class Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part A Part B
d(sin x) = cos x dxd(cos x) = sin x dx
y = 1 sin x cos x
derivative practice
Quotient Rule
,
2.3 Product and Quotient Rules of Differentiation
Calculus Home Page
Class Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part A Part B
one way to find the derivative, though not the easiest
derivative practice
Quotient Rule
Product and Quotient Rules of Differentiation
© G. Battaly 2015 16
July 03, 2015
Quotient Rule2.3 Product and Quotient Rules of Differentiation
Calculus Home Page
Class Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part A Part B
easier to use trig identities
G: y = sec x tan x F: dy/dx
dy = sec x tan x sec2xdx
y = 1 sin x cos x cos x
2.3 Product and Quotient Rules of Differentiation
Calculus Home Page
Class Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part A Part B
d(csc x) = csc x cot x dx
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
co‐functions
Product and Quotient Rules of Differentiation
© G. Battaly 2015 17
July 03, 2015
2.3 Product and Quotient Rules of Differentiation
Calculus Home Page
Class Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part A Part B
derivative practice
2.3 Product and Quotient Rules of Differentiation
Calculus Home Page
Class Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part A Part B
derivative practice
Product and Quotient Rules of Differentiation
© G. Battaly 2015 18
July 03, 2015
Calculus Home Page
Class Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part A Part B
derivative practice2.3 Product and Quotient Rules of Differentiation
2.3 Product and Quotient Rules of Differentiation
Calculus Home Page
Class Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part A Part B
derivative practice
Product and Quotient Rules of Differentiation
© G. Battaly 2015 19
July 03, 2015
Calculus Home Page
Class Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part A Part B
Calculus Home Page
Class Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part A Part B
derivative practice
Product and Quotient Rules of Differentiation
© G. Battaly 2015 20
July 03, 2015
Calculus Home Page
Class Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part A Part B
2.3 Product and Quotient Rules of Differentiation
Calculus Home Page
Class Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part A Part B
derivative practiceQuotient Rule
Product and Quotient Rules of Differentiation
© G. Battaly 2015 21
July 03, 2015
Calculus Home Page
Class Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part A Part B
Quotient Rule
Calculus Home Page
Class Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part A Part B
derivative practice
Product and Quotient Rules of Differentiation
© G. Battaly 2015 22
July 03, 2015
Calculus Home Page
Class Notes: Prof. G. Battaly, Westchester Community College, NY
Homework Part A Part B
