﻿

devexpress asp.net pdf viewer : Convert selected pages of pdf to word online application control tool html web page .net online Calculus6-part766

3.3 TheProductRule
61
g
0
(x)must actuallyexistforthistomakesense. . Wealsoreplaced lim
x!0
f(x+x)with
f(x)|whyisthisjustied?
Whatwereallyneedtoknowhereisthat lim
x!0
f(x+x)=f(x),orinthelanguage
0
(x)exists(orthewhole
approach,writing thederivativeof fg intermsoff
0
and g
0
,doesn’t make sense). This
turnsouttoimplythatf iscontinuousaswell. Here’swhy:
lim
x!0
f(x+x)= lim
x!0
(f(x+x) f(x)+f(x))
= lim
x!0
f(x+x) f(x)
x
x+ lim
x!0
f(x)
=f
0
(x)0+f(x)=f(x)
Tosummarize:theproductrulesaysthat
d
dx
(f(x)g(x))=f(x)g
0
(x)+f
0
(x)g(x):
Returningtotheexamplewestartedwith,letf(x)=(x
2
+1)(x
3
3x).Thenf
0
(x)=
(x
2
+1)(3x
2
3)+(2x)(x
3
3x)=3x
4
3x
2
+3x
2
3+2x
4
6x
2
=5x
4
6x
2
3,
asbefore. Inthiscaseitisprobablysimplertomultiplyf(x)outrst,thencomputethe
derivative;here’sanexampleforwhichwereallyneedtheproductrule.
EXAMPLE 3.3.1
Compute thederivative off(x)=x
2
p
computed
d
dx
p
625 x=
x
p
625 x2
.Now
f
0
(x)=x
2
x
p
625 x2
+2x
p
625 x2=
x
3
+2x(625 x
2
)
p
625 x2
=
3x
3
+1250x
p
625 x2
:
Exercises3.3.
In1{4,ndthederivativesofthefunctionsusingtheproductrule.
1. x
3
(x
3
5x+10))
2. (x
2
+5x 3)(x
5
6x
3
+3x
2
7x+1))
3.
p
x
p
625 x)
4.
p
625 x2
x20
)
5. Usethe e product rule to computethe derivativeoff(x)= (2x 3)
2
. Sketchthefunction.
Findanequationofthetangentlinetothecurveatx=2.Sketchthetangentlineatx=2.
)
Convert selected pages of pdf to word online - copy, paste, cut PDF pages in C#.net, ASP.NET, MVC, Ajax, WinForms, WPF
Easy to Use C# Code to Extract PDF Pages, Copy Pages from One PDF File and Paste into Others
extract pages from pdf online tool; cut paste pdf pages
Convert selected pages of pdf to word online - VB.NET PDF Page Extract Library: copy, paste, cut PDF pages in vb.net, ASP.NET, MVC, Ajax, WinForms, WPF
Detailed VB.NET Guide for Extracting Pages from Microsoft PDF Doc
copying a pdf page into word; copy pages from pdf to another pdf
62
Chapter 3 3 Rulesfor r FindingDerivatives
6. Supposethatf,g,andharedierentiablefunctions.Showthat(fgh)
0
(x)=f
0
(x)g(x)h(x)+
f(x)g
0
(x)h(x)+f(x)g(x)h
0
(x).
7. Stateandprovearuletocompute(fghi)
0
(x),similartotheruleinthepreviousproblem.
Productnotation. Supposef
1
;f
2
;:::f
n
arefunctions. Theproductofallthesefunctions
canbewritten
Yn
k=1
f
k
:
Thisissimilartotheuseof
X
todenoteasum.Forexample,
Y5
k=1
f
k
=f
1
f
2
f
3
f
4
f
5
and
Yn
k=1
k=12:::n=n!:
Wesometimesusesomewhatmorecomplicatedconditions;forexample
Yn
k=1;k6=j
f
k
denotestheproductoff
1
throughf
n
exceptforf
j
. Forexample,
Y5
k=1;k6=4
x
k
=xx
2
x
3
x
5
=x
11
:
8. Thegeneralized d product rulesaysthatiff
1
;f
2
;:::;f
n
aredierentiablefunctions atx
then
d
dx
Yn
k=1
f
k
(x)=
Xn
j=1
0
@
f
0
j
(x)
Yn
k=1;k6=j
f
k
(x)
1
A
:
outwhatthissayswhenn=5.
Whatisthederivativeof(x
2
+1)=(x
3
3x)? Moregenerally,we’dliketohaveaformula
to compute the derivative off(x)=g(x) if we alreadyknowf
0
(x) and g
0
f(x)=g(x)=f(x)(1=g(x)),thatis,thisis\really"aproduct,andwecancomputethe
derivativeifweknowf
0
(x)and(1=g(x))
0
. Soreallythe e onlynewbitofinformationwe
needis(1=g(x))intermsofg0(x). Aswiththeproductrule,let’ssetthisupandseehow
C# PDF Page Insert Library: insert pages into PDF file in C#.net
page2 }; // Specify a position for inserting the selected pages. doc2.InsertPages( pages, pageIndex); // Output the new document Insert Blank Page to PDF File in
delete blank pages from pdf file; copy one page of pdf to another pdf
VB.NET PDF Page Insert Library: insert pages into PDF file in vb.
page2} ' Specify a position for inserting the selected pages. doc2.InsertPages(pages, pageIndex) ' Output the new document. Insert Blank Page to PDF File Using
cut paste pdf pages; delete blank pages from pdf file
3.4 TheQuotient t Rule
63
farwecanget:
d
dx
1
g(x)
= lim
x!0
1
g(x+x)
1
g(x)
x
= lim
x!0
g(x) g(x+x)
g(x+x)g(x)
x
= lim
x!0
g(x) g(x+x)
g(x+x)g(x)x
= lim
x!0
g(x+x) g(x)
x
1
g(x+x)g(x)
g0(x)
g(x)2
Nowwecanputthistogetherwiththeproductrule:
d
dx
f(x)
g(x)
=f(x)
g
0
(x)
g(x)2
+f
0
(x)
1
g(x)
=
f(x)g
0
(x)+f
0
(x)g(x)
g(x)2
=
f
0
(x)g(x) f(x)g
0
(x)
g(x)2
:
EXAMPLE3.4.1
Computethederivativeof(x
2
+1)=(x
3
3x).
d
dx
x
2
+1
x3 3x
=
2x(x
3
3x) (x
2
+1)(3x
2
3)
(x3 3x)2
=
x
4
6x
2
+3
(x3 3x)2
:
seen. Sinceeveryquotientcan n be written asa product,itisalwayspossible tousethe
productruletocomputethederivative,thoughitisnotalwayssimpler.
EXAMPLE3.4.2 Findthederivativeof
p
625 x2=
p
xintwoways:usingthequotient
rule,andusingtheproductrule.
Quotientrule:
d
dx
p
625 x2
p
x
=
p
x( x=
p
625 x2)
p
625 x21=(2
p
x)
x
:
Notethatwehaveused
p
x=x
1=2
tocomputethederivativeof
p
xbythepowerrule.
Productrule:
d
dx
p
625 x2x
1=2
=
p
625 x2
1
2
x
3=2
+
x
p
625 x2
x
1=2
:
Withabitofalgebra,bothofthesesimplifyto
x2+625
2
p
625 x2x3=2
:
VB.NET PDF - Annotate PDF Online with VB.NET HTML5 PDF Viewer
Support to insert note annotation to replace PDF text. Ability to insert a text note after selected text. VB.NET HTML5 PDF Viewer: Annotate PDF Online.
extract pdf pages reader; delete page from pdf
C# HTML5 PDF Viewer SDK to annotate PDF document online in C#.NET
Support to insert note annotation to replace PDF text. Ability to insert a text note after selected text. C# HTML5 PDF Viewer: Annotate PDF Online.
extract pages from pdf without acrobat; combine pages of pdf documents into one
64
Chapter 3 3 Rulesfor r FindingDerivatives
Occasionallyyou willneed to compute the derivative of aquotient witha constant
numerator,like10=x
2
. Ofcourseyoucanusethequotientrule,butitisusuallynotthe
easiestmethod. Ifwedouseithere,weget
d
dx
10
x2
=
x
2
0 102x
x4
=
20
x3
;
sincethederivativeof10is0.Butitissimplertodothis:
d
dx
10
x2
=
d
dx
10x
2
= 20x
3
:
2
isaparticularlysimpledenominator,butwewillseethatasimilarcalcu-
lationisusuallypossible. Anotherapproachistorememberthat
d
dx
1
g(x)
=
g
0
(x)
g(x)2
;
butthisrequiresextramemorization. Usingthisformula,
d
dx
10
x2
=10
2x
x4
:
Notethatwerstuselinearityofthederivativetopullthe10outinfront.
Exercises 3.4.
Findthederivativesofthefunctionsin1{4usingthequotientrule.
1.
x
3
x 5x+10
)
2.
x
2
+5x 3
x 6x+3x 7x+1
)
3.
p
x
p
625 x2
)
4.
p
625 x2
x20
)
5. Findanequationforthetangentlinetof(x)=(x
2
4)=(5 x)atx=3. )
6. Findanequationforthetangentlinetof(x)=(x 2)=(x
3
+4x 1)atx=1.)
7. LetP P beapolynomialofdegreenandletQbeapolynomialofdegreem(withQnotthe
zeropolynomial).Usingsigmanotationwecanwrite
P=
Xn
k=0
a
k
x
k
;
Q=
Xm
k=0
b
k
x
k
:
UsesigmanotationtowritethederivativeoftherationalfunctionP=Q.
8. Thecurvey=1=(1+x
2
)isanexampleofaclassofcurveseachofwhichiscalledawitch
of Agnesi. . Sketchthe e curve andndthetangent linetothecurve at x=5. . (Theword
VB.NET PDF Image Redact Library: redact selected PDF images in vb.
Text. PDF Export. Convert PDF to Word (.docx). Convert to Text. Convert PDF to JPEG. Convert PDF to Png Page: Create Thumbnails. Page: Insert PDF Pages. Page: Delete
extract pages from pdf files; extract pages from pdf file
C# Word - Insert Blank Word Page in C#.NET
doc1 = new DOCXDocument(inputFilePath1); // Specify a position for inserting the selected page. Add and Insert Multiple Word Pages to Word Document Using C#.
crop all pages of pdf; delete page from pdf document
3.5 The e ChainRule
65
witch hereisamistranslationoftheoriginalItalian,asdescribedat
http://mathworld.wolfram.com/WitchofAgnesi.html
and
http://instructional1.calstatela.edu/sgray/Agnesi/
WitchHistory/Historynamewitch.html.)
)
9. Iff
0
(4)=5,g
0
(4)=12,(fg)(4)=f(4)g(4)=2,andg(4)=6,computef(4)and
d
dx
f
g
at4.
)
So far we have seen how w to o compute the e derivative e of f a a function n built t up p from m other
functions by addition, , subtraction, , multiplication and division. There e is another very
important way y that t we combine simple functions s to o make more complicated functions:
functioncomposition,asdiscussedinsection2.3. Forexample,consider
p
625 x2.This
functionhasmanysimplercomponents,like625andx
2
,andthenthereisthatsquareroot
symbol,sothesquarerootfunction
p
x=x
1=2
isinvolved. Theobviousquestionis: : can
wecomputethederivativeusingthederivativesoftheconstituents625 x
2
and
p
x?We
canindeed. Ingeneral,iff(x)andg(x)arefunctions,wecancomputethederivativesof
f(g(x))andg(f(x))intermsoff
0
(x)andg
0
(x).
EXAMPLE 3.5.1
Form the e two o possible e compositions of f(x) ) =
p
x and d g(x) ) =
625 x
2
and compute the derivatives. First, , f(g(x)) ) =
p
625 x2,and d thederivative
is x=
p
625 xaswehaveseen.Second,g(f(x))=625 (
p
x)
2
=625 xwithderiva-
tive  1. Of f course, , these e calculations s do o not use e anything g new, , and in particularthe
derivativeoff(g(x))wassomewhattedioustocomputefromthedenition.
Supposewewantthederivativeoff(g(x)).Again,let’ssetupthederivativeandplay
somealgebraictricks:
d
dx
f(g(x))= lim
x!0
f(g(x+x)) f(g(x))
x
= lim
x!0
f(g(x+x)) f(g(x))
g(x+x)) g(x)
g(x+x)) g(x)
x
Nowweseeimmediatelythatthesecondfractionturnsintog
0
(x)whenwetakethelimit.
denominator,g(x+x)) g(x),isa a change in the value of g,so let’sabbreviate it as
C# PDF Page Rotate Library: rotate PDF page permanently in C#.net
Online C# class source codes enable the ability to rotate int rotateInDegree = 270; // Rotate the selected page. C#.NET Demo Code to Rotate All PDF Pages in C#
cut pages from pdf file; extract pages from pdf
VB.NET PDF insert image library: insert images into PDF in vb.net
Access to freeware download and online VB.NET class to provide users the most individualized PDF page image as Png, Gif and TIFF, to any selected PDF page with
66
Chapter 3 3 Rulesfor r FindingDerivatives
g=g(x+x)) g(x),whichalsomeansg(x+x)=g(x)+g.Thisgivesus
lim
x!0
f(g(x)+g) f(g(x))
g
:
Asxgoesto0,itisalsotruethatggoesto0,becauseg(x+x)goestog(x). Sowe
canrewritethislimitas
lim
g!0
f(g(x)+g) f(g(x))
g
:
0
(g(x)),thatis,thefunctionf
0
(x)with
xreplacedbyg(x).Ifthisallwithstandsscrutiny,wethenget
d
dx
f(g(x))=f
0
(g(x))g
0
(x):
Unfortunately,thereisasmall awintheargument.Recallthatwhatwemeanbylim
x!0
involveswhathappenswhenxiscloseto0butnotequalto0. Thequalicationisvery
important,sincewemustbeabletodividebyx. Butwhenxiscloseto0butnotequal
to0,g=g(x+x)) g(x)iscloseto0andpossiblyequalto0. . Thismeansitdoesn’t
reallymakesensetodivide byg. Fortunately,it t ispossible torecasttheargument to
avoidthisdiculty,but it isa bit tricky; ; wewillnot t includethe details,whichcan be
found inmanycalculusbooks. Note e that manyfunctions s g do o have the property y that
g(x+x) g(x)6=0whenxissmall,andforthese e functionsthe argumentaboveis
ne.
ThechainrulehasaparticularlysimpleexpressionifweusetheLeibniznotationfor
thederivative. Thequantityf0(g(x))isthederivativeoff withxreplacedbyg;thiscan
bewrittendf=dg.Asusual,g
0
(x)=dg=dx. Thenthechainrulebecomes
df
dx
=
df
dg
dg
dx
:
Thislooksliketrivialarithmetic,butitisnot:dg=dxisnotafraction,thatis,notliteral
division,butasinglesymbolthatmeansg
0
(x).Nevertheless,itturnsoutthatwhatlooks
liketrivialarithmetic,andisthereforeeasytoremember,isreallytrue.
It willtakeabit ofpracticetomake theuseofthe chainrulecomenaturally|itis
morecomplicatedthantheearlierdierentiationruleswehaveseen.
EXAMPLE 3.5.2
Compute thederivativeof
p
p
625 x2,computeddirectlyfromthelimit. Inthecontext t ofthechain
rule, we e have f(x) ) =
p
x, g(x) ) = 625 x
2
. We e know that f
0
(x) = (1=2)x
1=2
, so
C# WPF PDF Viewer SDK to annotate PDF document in C#.NET
1. Highlight text. Click to highlight selected PDF text content. 2. Underline text. Click to underline selected PDF text. 3. Wavy underline text.
copy one page of pdf; delete pages out of a pdf
VB.NET PDF - Annotate PDF with WPF PDF Viewer for VB.NET
1. Highlight text. Click to highlight selected PDF text content. 2. Underline text. Click to underline selected PDF text. 3. Wavy underline text.
export one page of pdf preview; delete page from pdf preview
3.5 The e ChainRule
67
f
0
(g(x))=(1=2)(625 x
2
)
1=2
. Notethatthisisatwostepcomputation: rstcompute
f
0
(x),thenreplacexbyg(x). Sinceg
0
(x)= 2xwehave
f
0
(g(x))g
0
(x)=
1
2
p
625 x2
( 2x)=
x
p
625 x2
:
EXAMPLE3.5.3
Computethederivativeof1=
p
625 x2. Thisisaquotientwitha
constantnumerator,sowecouldusethequotientrule,butitissimplertousethechain
rule.Thefunctionis(625 x
2
)
1=2
,thecompositionoff(x)=x
1=2
andg(x)=625 x
2
.
Wecomputef
0
(x)=( 1=2)x
3=2
usingthepowerrule,andthen
f
0
(g(x))g
0
(x)=
1
2(625 x2)3=2
( 2x)=
x
(625 x2)3=2
:
Inpractice,ofcourse,youwillneedtousemorethanoneoftheruleswehavedeveloped
tocomputethederivativeofacomplicatedfunction.
EXAMPLE3.5.4
Computethederivativeof
f(x)=
x
2
1
x
p
x2+1
:
The\last"operationhere isdivision,sotoget startedweneedtousethe quotient rule
rst.Thisgives
f
0
(x)=
(x2 1)0x
p
x2+1 (x2 1)(x
p
x2+1)0
x2(x2+1)
=
2x2
p
x2+1 (x2 1)(x
p
x2+1)0
x2(x2+1)
:
Nowweneed tocomputethe derivative ofx
p
x2+1. Thisisaproduct,so o weusethe
productrule:
d
dx
x
p
x2+1=x
d
dx
p
x2+1+
p
x2+1:
Finally,weusethechainrule:
d
dx
p
x2+1=
d
dx
(x
2
+1)
1=2
=
1
2
(x
2
+1)
1=2
(2x)=
x
p
x2+1
: