product and quotient rules of differentiation · product and quotient rules of differentiation ©...
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![Page 1: Product and Quotient Rules of Differentiation · 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,](https://reader030.vdocuments.mx/reader030/viewer/2022041106/5f07a1877e708231d41df2b0/html5/thumbnails/1.jpg)
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
![Page 2: Product and Quotient Rules of Differentiation · 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,](https://reader030.vdocuments.mx/reader030/viewer/2022041106/5f07a1877e708231d41df2b0/html5/thumbnails/2.jpg)
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
![Page 3: Product and Quotient Rules of Differentiation · 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,](https://reader030.vdocuments.mx/reader030/viewer/2022041106/5f07a1877e708231d41df2b0/html5/thumbnails/3.jpg)
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
![Page 4: Product and Quotient Rules of Differentiation · 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,](https://reader030.vdocuments.mx/reader030/viewer/2022041106/5f07a1877e708231d41df2b0/html5/thumbnails/4.jpg)
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
![Page 5: Product and Quotient Rules of Differentiation · 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,](https://reader030.vdocuments.mx/reader030/viewer/2022041106/5f07a1877e708231d41df2b0/html5/thumbnails/5.jpg)
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
![Page 6: Product and Quotient Rules of Differentiation · 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,](https://reader030.vdocuments.mx/reader030/viewer/2022041106/5f07a1877e708231d41df2b0/html5/thumbnails/6.jpg)
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
![Page 7: Product and Quotient Rules of Differentiation · 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,](https://reader030.vdocuments.mx/reader030/viewer/2022041106/5f07a1877e708231d41df2b0/html5/thumbnails/7.jpg)
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
![Page 8: Product and Quotient Rules of Differentiation · 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,](https://reader030.vdocuments.mx/reader030/viewer/2022041106/5f07a1877e708231d41df2b0/html5/thumbnails/8.jpg)
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
![Page 9: Product and Quotient Rules of Differentiation · 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,](https://reader030.vdocuments.mx/reader030/viewer/2022041106/5f07a1877e708231d41df2b0/html5/thumbnails/9.jpg)
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
![Page 10: Product and Quotient Rules of Differentiation · 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,](https://reader030.vdocuments.mx/reader030/viewer/2022041106/5f07a1877e708231d41df2b0/html5/thumbnails/10.jpg)
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
![Page 11: Product and Quotient Rules of Differentiation · 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,](https://reader030.vdocuments.mx/reader030/viewer/2022041106/5f07a1877e708231d41df2b0/html5/thumbnails/11.jpg)
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
![Page 12: Product and Quotient Rules of Differentiation · 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,](https://reader030.vdocuments.mx/reader030/viewer/2022041106/5f07a1877e708231d41df2b0/html5/thumbnails/12.jpg)
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
![Page 13: Product and Quotient Rules of Differentiation · 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,](https://reader030.vdocuments.mx/reader030/viewer/2022041106/5f07a1877e708231d41df2b0/html5/thumbnails/13.jpg)
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
![Page 14: Product and Quotient Rules of Differentiation · 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,](https://reader030.vdocuments.mx/reader030/viewer/2022041106/5f07a1877e708231d41df2b0/html5/thumbnails/14.jpg)
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
![Page 15: Product and Quotient Rules of Differentiation · 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,](https://reader030.vdocuments.mx/reader030/viewer/2022041106/5f07a1877e708231d41df2b0/html5/thumbnails/15.jpg)
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
![Page 16: Product and Quotient Rules of Differentiation · 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,](https://reader030.vdocuments.mx/reader030/viewer/2022041106/5f07a1877e708231d41df2b0/html5/thumbnails/16.jpg)
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
![Page 17: Product and Quotient Rules of Differentiation · 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,](https://reader030.vdocuments.mx/reader030/viewer/2022041106/5f07a1877e708231d41df2b0/html5/thumbnails/17.jpg)
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
![Page 18: Product and Quotient Rules of Differentiation · 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,](https://reader030.vdocuments.mx/reader030/viewer/2022041106/5f07a1877e708231d41df2b0/html5/thumbnails/18.jpg)
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
![Page 19: Product and Quotient Rules of Differentiation · 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,](https://reader030.vdocuments.mx/reader030/viewer/2022041106/5f07a1877e708231d41df2b0/html5/thumbnails/19.jpg)
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
![Page 20: Product and Quotient Rules of Differentiation · 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,](https://reader030.vdocuments.mx/reader030/viewer/2022041106/5f07a1877e708231d41df2b0/html5/thumbnails/20.jpg)
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
![Page 21: Product and Quotient Rules of Differentiation · 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,](https://reader030.vdocuments.mx/reader030/viewer/2022041106/5f07a1877e708231d41df2b0/html5/thumbnails/21.jpg)
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
![Page 22: Product and Quotient Rules of Differentiation · 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,](https://reader030.vdocuments.mx/reader030/viewer/2022041106/5f07a1877e708231d41df2b0/html5/thumbnails/22.jpg)
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