12YEARSOF MATH

MATHIS ALANGUAGE

1 Somanysymbols
Whyarethere somanysymbolsinmath
Take 2 3 5 Howdowereadthis twoplusthreeisequalto five
Nowimagine writingthis a 100times Writinginwordstakesso muchtimeandeffort

Howcanwe savetimeandeffort introducesymbols

Conclusion mathsymbols are alphabets or letters ofthe languagecalled mathematics

Anyquestions sofar
2 Mathematics is a language oflogic

Ever difficultwords

9 I
English hungry vs antepenultimate

Math us fix

Asyoulearn a languagemoreandmore youstart to understandthecultureofthelanguagemoreandmore

Asyoulearnmathmoreandmore youstart tounderstandlogicmoreandmore
3 Howtogetbetter at a language
learnwords learnsymbols
learngrammar learnhowto usesymbols

practiceusingthelanguage practice mathematics

Youhavedonelotsandlotsof butnotenoughof or
EQUALITY
LEARN
Fgm
meansequal whatis writtenontheleftis equf.iniisinti iifijiived
If twoobjects wereequal doingthesame operation to both objectskeepthingsequal
understandingthisrequireslogic does
thatmakesense
Example Imagine aboxofgrapes Toweighthegrapes we useoldfashionedscale

edited
ighgrapes
go.it
aierthanlkg 000h.It.sk sayssame.fi
go.ofcoursei
go.la go
If we writetheabovesituation usingthelanguagecalled Iinathematics
x 0 1 2 3 22 3 2

Example twofunctions ofthreevariables fixingz gosling z

If then
ff wedidn'tlearnthis
fixingz DON'Tworry
glory ffcx.mg

Wecan'tread understand allthesymbols

whatiswritenontheleftisequaltowhatiswritenonhen.gl
butweknow hemeaningof g
Itwbeatsereequaloinghesameperationtobotojectsleephingseget
thefirstlogicalrule

fixing2 gloryZ
whatiswritten theleft what writtenontheright

of
opteration
ix
g offguyz
sameoperation

Example Let's
say x ytz Then xt3 ytzt3
addition

x Z
y z Z subtraction
y
4x 4 ytz 4g42 multiplication

X Ly Z 1212 division

xcytz y 2yz z exponentiation

Fc set Cytz FZ

log log Cytz thelogarithm

taking

fca fcytz evaluation of afunction

SECONDLOGICTOLEARN INVERSE

needtochoose afont
Imagine
youareusingphotoshopillustratorto design a posterand
selectalltextsandthenchangethefont butyoulikethefirstonebetter
sowe select UNDO or If
firstfont font
second

Thisis aprocessthat happens a lotin reallife

objectsituation changedobjectsituation

undo
maynotbe availableformanyoperations
Inmaththisis a CENTRAL concept
manyoperations actionscomewith undo or

inverseoperations actions

Wealreadyknowmanyexamplesusing our logic

operation inverseoperation

2 3 addition x 3 subtraction

É 2x multiplication division

x exponentiation I
4 log X
sin x sin se

Fromthislistweseethat

inverseoperation is ALWAYS harder counterintuitive 411011 51E

Whenwefirstlearnmath westartwithnaturalnumbersand
addition

24 6 43 608 651

Butwhenwelearnsubtraction with natural numbers

6 2 4 651 608 43 2 6 4 weneednegativeintegers

Sowelearnintegers andmoveon to multiplication

2 3 6 447 3 12
 Butwhenwelearndivisionwithintegers
6 3 2 C12 3 4 3 4 weneed rationalnumbers

Inthesamewaylearningsquarerootrequiresus to learn irrationalnumbers ex T2

Whyaresquarerootsandlogarithms so difficult Becausetheyare counterintuitive especially

ex1 Howwouldyouread Is Whatnumberis equalto squarerootof 25

whatpositivenumbersquared is 25
25

exz Howwouldyouread log16 Whatnumber is equalto log4 of 16

16 4raisedtowhatpoweris 16

So ft isthenumber whensquared isequalto 7 F 7

Andlog7 isthepoweryouneedto gofrom 3 7 310937
7

LEYTO SOLVINGEQUATIONS EQUALITY INVERSE LOGIC

Let'ssolveequationsusinglogic

e1 If 2 6 4 find x
2 6 4 2 6 4
6 6 x2
2x 4 6 10 1 3 164 2
3
2
x 12110 52 3 x 2 3 5
e2 If 2 6 4 find4k
2 6 4
69 6
4 6 10
1 2
42 21 10 20
a3 If 3 4 7 find
3 4 7
317 log
x 4 10937
4 4
log7 4
Someequationsarenotso straightforward to seethe
steps
ex4 I asitbutc o find x

19

Since has si perhaps weshouldsquarebothsides butwealsohave x in

Trickisto addanumberandthen squaring

N
N
NJ I 2Nx N

Thencomparing x̅ with asi wecanguess astep multiplybya

a
it EEI
Soweneedtomake it bax looklike it 2Nx N
Compare and toget 2N
my
N
IN E
Wehavealltheingredientsnow
ax bxtc 0
af I
xta.gg
52 8 0

x x dg
xtzaj sitfxtfa.fm 14 2

a
I
Eat x b 4 1 Ea
2
 Atthispoint youmighthavequestions
howdoI come withthis
up
AnsYoudon'thaveto It istheteacher'sjob
howdo I rememberthis

AnsYoucanonlyrememberthis withoutmemorizing bymakingthisknowledgeyours

I rememberthis because I madethisknowledgemine
Forexample I solve set4 1 0 inthefollowing way
54 1 0
5 24 5
4 5
21 2 155
I r
2 2
x 2155
hesetworulesoflogic is keyto understandingthebigpicture especiallywhencombinedwiththeconcept offunctions
ENTRALCONCEPT FUNCTIONS

Myfavoriteexample of functions machines

getfriedpotato
CARROTS
iif
itff.it iii iiiis
When I
putcarrot intheFryer I getfriedcarrot
f

iig
friedcarrot

qg
Fryercarrot
When I
putfish intheFryer I getfriedfish
Fryerfish friedfish
FRYER hasaninput and an output FUNCTION almost

Q Can I putwaterin
youcan'tputinthefryer
No Thereare somethings

So weneedto makesurewhatwe canputinthe fryer DOMAIN 2104

Allthepossiblethingsthefryercanmake RANGE 19
Q Can I getanythingotherthanfriedpotatowhen I putpotatointhefryer NO

fryerpotato friedpotato
always

anelementintheDomain is relatedto
onlyoneelementintheRANGE
Q Can Igetfriedpotato if I putin somethingelse YES
geritroenpotato fried

morethanoneelementinthepoman canbe

relatedtothesameelementintheRANGE
fffff
f ftp.t

Let'sMIXFIRST SECONDRULE
OFLOGICTO FUNCTIONS
Rute Equlit If potato fryerpotato
Egg
fryer d
fryers fryerpotato friedpotato
 Rule 2 inverse Canweundofunctions

potato

fryera fryerpotato friedpotato

Whatdowe
putinthefryerto get friedpotato fryer friedpotato
Ans potato frozenpotato fryer friedpotato
Wecanundofunctions butthisinverseoperation isNOT a function
MATHEMATICALFUNCTIONS

1 Characteristics

hasinputandoutput
DOMAIN allpossibleinputs
RANGE allpossibleoutputs atdefinition offunction

oneinput Itodifferentoutputs oneinput one poggge.mg

twoinput oneoutput

2 Nomenclature

functionname usually aletter fg

input x
fix
Output
y
Example Let for of tankx fsncu.ms du
Then f 2 22tanh2 f sncu.msdu
flytz cytz tanhcytz
f sncu.msdu

f peanut peanut tankpeanut f cu.msdu

f A tanh f sncu.ms du
3 Logicandfunctions

Reequality 54 212
onepossibleoutpnffffa.se
oneinput

for
NOTEthisruleispossiblebecauseofthedefinition offunctions
 Ruletiverse dz
maynotbe a
f function
fan fu
Example Forfew x finda suchthat f 25 x 5
So fundo25 5 fundo not afunction

Can undoing afunction beagain a function

Example For for 2 3 findsosuchthat fish 5

So fund 5 1 and it is okfor 5

Butwhat if there issome y where fundly multipleanswers

Let'scheck
3
IS 2
Icy 3
so forany thereis
onlyone x suchthat fix
y y
so forany thereis
onlyone x suchthat funding x
y
fundo
is a function
Inthiscase fund iscalledtheinversefunction of f f21 4 andwrite f
4 Graph of a function
Wecanvisualizeanyfunction using a graph

Ex for 2 3
glad FE
has x2 2
 Q Whyis thegraph of f
Whyis A thegraph of g
Whyis thegraph of h

Point a b is onthegraphof f fca b is true

Whatis agraphof afunction f

a collectionof allpoints x y suchthat fix y

If weunderstandwhat a graphis we canalsounderstandthefollowing

If ab is a solution to twofunctions f and g then a b is an intersection

ofthegraphsof f and
g
a b solutionto f fca b so ab is onthegraphof f
carbs solutionto
9 s onthegraphof 9

mu t mu
9,9
cab
Agraphof a function cannotintersect a verticallinemorethanonce
a verticalline collectionof allpointswhose x coordinateisthesamenumber C

equationof a verticalline x c

If agraphintersects a verticallinemorethanonce forexample two

thenthepointshavecoordinate C Y and c
Ya
Hence wehave flo y for y f is not afunction
verticallinetest

Graphofaninversefunction f andthegraphof f are symmetricwithrespect totheline x

y
a b IS
Eample
y fix 2 3
f a 2at3 b
e
2a b 3 P forthefunction f
a feb3 fy
outputforthefunction
f
 tofundocy Icy3 x frombefore
compare

y f x In
If cabs is onthegraph
for outputforf
input off ftp.fffihesraphoft

f hasnoinversefunction ifthegraphof f intersect a horizontal linemorethanonce

ahorizontalline collection of allpointswhose coordinate isthesamenumber C

y
equationof a horizontalline y c

If agraphintersects ahorizontallinemorethanonce forexample two

thenthepointshavecoordinate x C and a c
Hence wehave flow c f x C andfundoc x andx2 fundo not afunction

horizontallinetest

thegraph
Analyzing
b
Whatisthegeneral shape

is zero
axis
the
Wheredoes
graphintersectthe y
findthepoint xoyo onthegraph off aid axis
findfco yo
 the
Wheredoes
graph intersectthe splintswhose coordinate
se is zero
equation y 0

findthepoint xoyo onthegraph offfind axis

find floco 0 UNDO

Example Linearfunction fin math

slope m
shape line

yintercept yo fco a b É c
Scintercept ax'tbx c QUADRATICEQUATION

Example Quadratic function fin ax'tbutc

shapeparabola

yintercept yo fco a b É c
Scintercept ax'tbx c QUADRATICEQUATION

Pattern function for

r
shape

g intercept set
c intercept
vaTIoN
fiiij
trigonometric 7

onclusion our ULTIMATEGOAL is to learn WHATFUNCTIONSARE 44 997T

Welearnsomanytypesoffunctionsbecausetheyare EXAMPLES offunctionswithreallifeapplications

IEEE EERN E.IE IE WHAYE8EPEEYIs ARE

thispartismissing
nderstanding WHATFUNCTIONS ARE iscrucialbecause incalculus we areinterestedinthe thstep

Pattern function
r
for f
shape

yintercept yo jiffy
Scintercept
IÉTION
fix ME
exponential
logarithmic EI
trigonometric 7
tangentlineTIE DIFFERENTIATION IE
LOPEISVELOCITY

WewillnowstarttolearnHowtofindthetangentlineandWHYwewanttofindthetangentline
Example walkingdowntheroad tmaf

a
walkingat 0.5kmh

after1hour I walked 0.5km

velocity
after2hours I walked 1km y fois Ix
after 4hours I walked2km
input c futputing

walkingat 2kmh walkingat 3kmh

after1hour I walked 2km after1hour I walked 3km

after2hours I walked4km y fu 2x after2hours I walked6km y fois 3x
after 4hours I walked 8km after 4hours I walked12km
speed equation

0.5 y x

2 1 2 slopeoflinearfunction

3 y 34
so wehave SLOPE VELOCITY

Wedon'tevenneed toknowtheequation tofindoutwhowalked thefastest

fastest
 Howaboutnonconstantspeed

distance
g
Whenwerewemovingthefastest
b

Howdoweknowthat
thegraphisthe steepest
a b c sitime
sheisthehighest
Weareusingourlogicandintuitiontoanswerthisproblem
butwehave nomathematicalreason
problem wecanonlycalculateslopesforlines

IDEA usetangentlines

Slopeofthetangentline
close
really totheslopeofthecurve.IM

goodapfroximation ofthevelocity

wecancalulatethis if wecanfindthetangentline

FINDINGTHETANGENTLINE

Example carnavigation onlyinformation ithas is location buthowdoesitknowyourspeed

yourpositionevery 1
checks second

Between 1 1 and2 youtravel

sowecanfindyourspeed distangffeled

d
t
If wewantto updateourspeedmoreoften
checkmoreoften 0.5sec 0.1sec0.01sec
to
thesmallerthetimeinterval thebetter
et'sapplythisideato our problem Pee

Fiat time b a

speed distanf.IE taffaI slope

4 feb x

Remember smallinterval meansbetter

As bgetscloserto a

iffff.fr b b
linegetsclosertothetangentline

Thismeans as bgetscloserto a
theslopeofthelinegetscloserto theslopeofthetangentline
Q Howclosecan b getto a b a smaller is better

Whynot b a or b a o

problem slope
ff so the denominator becomes

Q Canwesolvethisproblem
weneedto learn LIMITS
IMITS
Suppose I draw a graphbuthide a smallportionof it

f
Canyouguesstheequationoffish
it lookslike aline withslope7 and yintercept 2
Somaybe y fix oct2

Let'suncover a littlemore
f

so it seems wewereright

hegraphwas

Answer fix
Originalguess gas oct2

12 undefined

2 2 4
92
Butwenotice fix

oct2 gas so wewereright

danceled
roblemis we x 2

Let'sunderstandhowcancelingworksin afraction
Example 9 32
Wewillstartfrom andmakeit
32.1 sinceanynonzeronumbermultiplied
by1 is itself
33 sinceanynonzeronumberdividedbyitself is 1

96

Thisgivesus a hinthowwe cancel

9.1

3
Or we canalsosee p
at b
6 aa Gb

yggfgq.gg

reciprocal
Inbothexplanations we are mdyyingbythe ofthenumberwe arecanceling
noingback or dividingbythenumber

fix whenwecancel
by 2 wearedividingbyso 2 Ah
Q Whatif x 2 Thenwe are dividingby0

oct2 Thus whenwecancel

got byso 2 we are assuming x 2
CONCLUSION fix gesi when x 2
So fiz undefined Butisthatit
Can'twestillfinda wayto
say f 2 4
withoutactuallysayingit

EA 0 999 0.9 1
welearnthisinschools but are wesurethisis true 0.999 isnever1 butequalto 1
0.999 is a waytosay 1 withoutactuallysayingit

thisispossiblebecause 0.999 is infinitelycloseto 7

Whatdowemean
by infinitelyclose
trichotomy If a barenumbers thenoneof by istrue
So 1 0999 or 1 0.999 SC 0

But 1 0.999 is smallerthananypositivenumberhoweversmall

Q whatnumber 20 butstillsmallerthananypositivenumber howeversmall
2 0

1 0.999 2 0

et ustake soinfinitelycloseto 2 butnotequalto 2 Wearesaying so2 withoutactuallysayingit

wecandivideby so 2 so fish gas

Then fix is infinitelycloseto 4 Wearesaying fiz 4 withoutactuallysayingit

As x approaches 2 fin approaches 4

ftp.fixs 4 thelimitof fishas x approaches 2 is 4

ACKTOTANGENTLINES

Ourgoal findtheslope ofthetangentlineto y fcx at sea

y
IDEA Choosesomeother b
Findtheslopeofthe linethrough afca and b f b

Thesmallerthedifference between a andb thebetter

b
say a b butthenwe divideby0
Problem wewantto

Isthere awayto say a b withoutactuallysayingit

LIMITS
slope ofthetangentlineto
y fcx at x a

fila
ftp.fff
xy fixs x x

3y
fcxs mxtn sc

ay
fixs x2 2

IT x fix
functionwith
tangentlinetothegraphof f at
slopeofthe

schangetheidea a littlebit

Ya Ya

slopeofthetangentlineto f at sea

time if fiif fi
fu
Lp
RIVATIVES

fish fastest
day wewillcalculate derivatives andhopetofigureoutsomerules

y f ld x
REVIEW

y fa x

y fu f cutfaces setse JOSEPH

y has floutgas You

y face JOSEPH

y fsd JOSEPH

You
y gas

fc fqtatdf.it f a _fqt.at zf
t

tryst
I 1 dayide
Hmo 2x da 2x

him ftp.hatdfjh

ᵗ
i
iiiiiiiiiiiiiii.it
i iiiiiiiiii.fi
ftp
iii
iiiiiiii
dffd fjng9atdI9I
ficsi false fix gca
1 2x
ficn fjy.IE e s ftp.f tdffe
Lp e

Iii ftp.IL an
L It Cities

it tfiiifif fffi
liff.fi
L
t
t.ci
ftp.fxc.it
Iii ffife iy
t.ci a

L
ficas fellas 2x

gatf.mg dsf9
Lp
ftp f d.s

L I

ff.fi
ffffffff
L II It

fix 12
 HOMEWORK 1

Use the definition of derivatives

f (x + dx) f (x)
f 0 (x) = lim
dx!0 dx
to calculate the derivatives of the following functions.

1. f (x) = 3x 8

2. f (x) = x3 3x
1
3. f (x) =
2x + 1
p
4. f (x) = x3 3x
2 HOMEWORK 1

Answers.
1. f (x) = 3x 8
f (x + dx) f (x)
f 0 (x) = lim
dx!0 dx
(3(x + dx) 8) (3x 8)
= lim
dx!0 dx
3x + 3 dx 8 3x + 8
= lim
dx!0 dx
3 dx
= lim
dx!0 dx
= lim 3
dx!0
=3

2. f (x) = x3 3x
f (x + dx) f (x)
f 0 (x) = lim
dx!0 dx
((x + dx)3 3(x + dx)) (x3 3x)
= lim
dx!0 dx
x3 + 3x2 dx + 3x dx2 + dx3 3x 3 dx x3 + 3x
= lim
dx!0 dx
2 2 3
3x dx + 3x dx + dx 3 dx
= lim
dx!0 dx
dx(3x + 3x dx + dx2 3)
2
= lim
dx!0 dx
= lim (3x + 3x dx + dx2 3)
2
dx!0
2
= 3x 3

1
3. f (x) = .
=
2x + 1
0 f (x + dx) f (x)
f (x) = lim
dx!0 dx
1 1
2(x+dx)+1 2x+1
= lim
✓ dx
dx!0
◆
1 1 1
= lim
dx!0 dx
⇣
2(x + dx) + 1 2x + 1 =
.
i
⌘- d ( 2(xtdx)
*
1 1 2x+1 1 2(x+dx)+1
= lim dx 2(x+dx)+1 · 2x+1 2x+1 · 2(x+dx)+1
dx!0
1
⇣
2x+1 2(x+dx)+1
a
x
⌘ * (2cxtdx2 ,
*

= lim dx (2(x+dx)+1)(2x+1) (2x+1)(2(x+dx)+1)

dx!0
✓ ◆
1 (2x + 1) (2(x + dx) + 1)
= lim
dx!0 dx
✓
(2(x + dx) + 1)(2x + 1)
◆
dx ( 24tdx
n
)+1 ) ( 2 x
+

1 2x + 1 2x 2 dx 1
= lim
dx!0 dx (2(x + dx) + 1)(2x + 1)
✓ ◆
1 2 dx
= lim
dx!0 dx (2(x + dx) + 1)(2x + 1)
 HOMEWORK 1 3

2
= lim
dx!0 (2(x + dx) + 1)(2x + 1)
2
=
(2x + 1)2

p
4. f (x) = x3 3x
f (x + dx) f (x)
f 0 (x) = lim
dx!0
p dx
p
(x + dx)3 3(x + dx) x3 3x
= lim
dx!0
p dx p
p p
3
(x+dx) 3(x+dx) x3 3x (x+dx)3 3(x+dx)+ x3 3x
= lim dx
· p 3
p
3
dx!0 (x+dx) 3(x+dx)+ x 3x
((x+dx)3 3(x+dx)) (x3 3x) 1
= lim dx
·p p
dx!0 (x+dx)3 3(x+dx)+ x3 3x

2 1
= (3x
,
3) · lim p p
dx!0 (x + dx)3 3(x + dx) + x3 3x
2
3x 3
= p
2 x3 3x
OREDERIVATIVES

focus x

f.cas FC

f.cn I
god F
fiw f.mg fjfe f fI
ftp.facxtd

L L

iiiiiii
sina.is
3i ms Edit
Lp fioEdi
ficx

feI
f c _ftp.fecxtd ghafyggutdsy.is
f.i EIF

iiiiiiiii.ie
i
iiiiiiiiiii
iiiii O t fi tfn
finsE fi0 f a fiio twafFo atfn
fiw.at 3a Za this Iff
UNPROBLEM

Let'ssayyouare Davidandneedto fight Goliath Yourweapon slingshot

Asyouspinyourslingshot it moves in a circle
When let thestonemovesin a directiontangenttothecircle
you go

You TH
yd
r

QWhenshould weletgo to hitGoliath

wetaketheupperhalf ofthe d
1 a
x y for
x2 go.gg y
1 ya
y r
g ad AtH
I g
A FindthetangentlineLiu to fix thatgoesthrough 3,0
fix oct
1

this y gas 2 127 if

tangentline y Low 2 ex a fear

L3 far 13 a F
Faz 13 a
g Faz
3a a
14mar
1 as a 3 a

1 3a
1 92
a
 derivative ofa
1,4
E 2x
3 3 2

pattern

POWERRULE INTEGERS

Xn noont

xn
d ftp.t
ÉkÉÉ
ÉÉ
ÉfÉf
jIIInIfff
E xnkdxk w
ftp

i ii cnn.m.LY
11
non1

youarenotusedto
summation not i thenyoucanwritethe summationout

xn
d ftp.t IuseBinomialtheorem
den
EmEI
andi di xnkdock an
ftp 8 1
xjd.it
x di I an2dgt xnkdock
ftp
2 t i sentdocke
fig Ida
9an1 I xn2dec xnkdocke

nona
combinedwiththe sumrule hasfastgen hem fin gca
thederivativeofany polynomialfunctions
wecaneasilyfind

fix 24 3 2 2
f x 24 32 2
3
4 3.2 0
4 3 bx
 HOMEWORK 2

Use various rules of derivatives to calculate the derivatives of the following functions.

1. f (x) = 3x 8

2. f (x) = x3 3x

3. f (x) = x sin x

4. f (x) = (x2 2x 4) sin x

2 HOMEWORK 2

Answers.
1. f (x) = 3x 8 '
'
0
f (x) = (3x 8) 0 f x= ( 34- 8 )
'
0
= (3x) (8)0
= 3 · (x)0 (8)0
=
(
3 π7 -
(8)
.

=3·1 0 " (8)

= 3 .
(x)
=3
0
=
3 .
-

2. f (x) = x3 3x
f 0 (x) = (x3 3x)0
:
= (x3 )0 (3x)0 3x 3

= 3x2 3 · (x)0
= 3x2 3·1
2
= 3x 3

3. f (x) = x · sin x inx ) ' =

CoSx

f 0 (x) = (x · sin x)0 '

, sinx + x ( sinx)
= (x)0 · sin x + x · (sin x)0 λ x)
.

= 1 · sin x + x · cos x
+
= sin x + x · cos x = .
sinx x cosx .

cosx
sinxt x
.

4. f (x) = (x2 2x 4) sin x :

f 0 (x) = ((x2 2x 4) sin x)0 '

= (x 2
2x 0
4) sin x + (x 2
2x 4)(sin x) 0 ( Cosx ) = -
sinx

= ((x2 )0 (2x)0 (4)0 ) sin x + (x2 2x 4) cos x

= (2x 2 · (x)0 0) sin x + (x2 2x 4) cos x
= (2x 2 · 1) sin x + (x2 2x 4) cos x
2
= (2x 2) sin x + (x 2x 4) cos x

: 24 4) Cosx
(x
-

'

( sinx)
=
cosx

Co a) sinx
$ = -
DERIVATIVERULES

siderates

uniting
c.fm c fleas Fh
fix gas fix cguns fin cgon fin g's
xn n xn foranyinteger
fish 3 4 34 x2

fa 34 x 2
10
34 x2
y
31 4 t x 2
3 43 2 3

12 3
DIFFERENT NAMESFORDERIVATIVES

Recall our firstexampleforlearningthemeaningofthetangentline

FromLecture04

Theideaforfindingtheslopeofthetangentline

f tangentline
y fca x a fca
Ya

a b
at 3
 Choose b closeto a
Findtheslopeofthelinebetweentwopoints afca and b f b

Letbgoreallycloseto a

Whatistheslopemeasuring

slope

It ismeasuringhowmuch ymovedfcato f b when x movedfrom a to b

it ismeasuring changein y intermsofchangein x oct05stE05011Eter y7 Eat 5171

Buying apples at 9 vs 5apples at 10 Whichismorepriceefficient

whatischangingprice
y
whatwearemeasuringpriceagainst numberofapples x

As ofappleschangedfrom 0 to 3 thepricechanged from 0 to 9

418
17 3 s
3perapple

As ofappleschangedfrom 0 to 5 thepricechanged from 0 to 10

48
17 54 5
2perapple

Aswemake changein x verysmall theslopebecomestheslopeofthetangentline

f
tangentline
y fca x a fca
Ya

t a 3
dec
Ta
dy

dx
small changein x
very
dy verysmall changein y
f x slopeofthe
tangentline rateofchange If 1 64
THREEDIFFERENTNAMESOFTHEDERIVATIVEOF
y fun
fix theslopeofthetangentlineto fix
If measuring thechangein
y intermsof changein x
If x measuring thechangein f intermsof changein x
Ca theslopeofthetangentlineto fix Josephsfavor.it

fem If a writingderivatives as afunction

fca a IfCa whenevaluatingderivatives at x a

Vowwe canview derivatives as approximations ofchange

f
tangentline
y fca x a fca
Ya

finds
I
a atdx
dsc small changein x
very
If small changein f
very

Q Howmuchdidf change as x changeddo

df fix doc but as docgetsreallycloseto 0 theapproximationgetsbetter
df f cadoc
Thenextrulesofdifferentiation are EXTREMELYIMPORTANT
andthisidea istheKEYto understanding it

fix x f x 2x df 2xdo

f areaof
thesquare df
df changein area ssmall wecanignore

does dxchangeinsidelength x.de a a a I

DERIVATIVEOFTHESINEFUNCTION

the
Using idea sin diff
Whatis sin0 It isthe height ofthepointontheunitcirclemadebytheangle 0
1 unitcircle

sino

10 1

so d fˢ measuresthe
changein height ofthepoint interms ofthechangeintheangle
1 unitcircle

1
Fo
is

p
doing
7
do
oct
" "

disino
8 do
7
sinx cosse
cos0
fˢ
thedefinition
Using

1 unitcircle si htano f c
10
ton θ
sino
sino 0 tano
10 1
isfofff.si o costo

1251 coso
tf.iq I ftp.sifsfigcoso ftp
figsif 1 11
 I I c

fine

fistdoÉt
fine fi
fine

d m.si sin
iai

ˢ
i
sinx fi figcos x ftp.sina
sinx.o cosx 1
cosx

Howaboutcosx

ORODUCTRULE whathappens if we multiplytwofunctionsandthendifferentiate

has fix g x
has 4td h

day finglas
fingfatdagut
fjyofextdasgcxtdso fasgcxtdztfcxsgcxtdas flng.us

fjyofextdasgcxtdso fcxsgcxtdg.si fixgaitda flongas

das foisgait
finsfait day for garda gas

Yg
feudal
f gotdoes fix gated gun
ftp.fextdjfw.fiogextda finefix ftp9cxtd 9
f ingas foisg a
PRODUCTRULE

fixgas fingers fix gex

 2
x ( osx
xsinx
.

+
2
 HOMEWORK 3

Use various rules of derivatives to calculate the derivatives of the following functions.

1. f (x) = sin(x3 )

σ
2. f (x) = (tan x)4
sin x - :
* - ' sinx
*
- 2x 3inx
3. f (x) = 2 where x 6= 0
xx
x
= . x
2

x
( ←

)
4. f (x) = (sin(cos x))2

t sx
= 4- .

- 2

gin☆
^

CosU .

3
2 3
3x .
(03 x.

cosxca ,
f( x) = n uR
2 3
sincios x] ) = V f ( x) √

= 3x ( oEx ,

4
2 .
tanx =
u .
f ( x) =
u
.

=
eKu
"

"

3
. sectx ×
4 tomx)
 Hu
f ( x) =
sin u . ∞
)
u =
x
,
= cos u .

2
2
HOMEWORK 3 : 3x .
cosu
= 3x

@
∴
.

=
x

Use various rules of derivatives to calculate the derivatives of the following functions.

1. f (x) = sin(x3 )

2. f (x) = (tan x)4

sin x
3. f (x) = where x 6= 0
x2

□
4. f (x) = (sin(cos x))2
2

u
=
( os x .
f ( x) =
sinu)

[ V = sin u .
fcx) =
√

= ! -

cose
9 in ,

.
↓

-
2V .
cosu
o
- sinx

= 2 .
sinu .
cosu -
Sinx

=
2 .
sin .
cosxo cos cosx
. -
sinx

= -
2 sin losx . .
( os .
cosx
.

sinx
2 HOMEWORK 3

Answers.
1. f (x) = sin(x3 )
du
Let u = x3 so that dx
= 3x2 . Then we have
3
0 d U= x :

f (x) = (sin(x3 ))
dx
d
=
dx
(sin u)
3x
= ?
du
= cos u ·
dx
= 3x cos(x3 ).
2
fcxl = sinu .

2. f (x) = (tan x)4

du
Let u = tan x so that dx
= sec2 x. Then, fEx]= cosce .

d
f 0 (x) = ((tan x)4 )
dx
d 4
= (u )
dx
du
= 4u3 ·
dx
= 4 tan3 x sec2 x
sin x
3. f (x) = 2 where x 6= 0.
x
Use the quotient rule:
(sin x)0 (x2 ) (sin x)(x2 )0
f 0 (x) =
(x2 )2
cos x(x2 ) sin x(2x)
=
x4
x cos x 2 sin x
=
x3
4. f (x) = (sin(cos x))2
Use chain rule multiple times: Let
u = cos x
v = sin(cos x) = sin u.
Then we have
du
= sin x,
dx
dv d du
= (sin u) = cos u · = cos(cos x) · sin x,
dx dx dx
Therefore,
d
f 0 (x) = (sin(cos x))2
dx
d 2
= (v )
dx
dv
= 2v ·
dx
= 2 sin(cos x) · cos(cos x) · sin x.
eviewn

Differentiation to
gettheslopeofthetangentline
bestapproximationofrateofchange as x changes a littlebithowmuch is f changing

4 fix
Approximation ofchangein f df fix doc
Rulesofdifferentiation sofar siderules

fungus fix gin

cfan c fix
Can non ne 2

sina cos X

ustfasgi
cnn.SI
i
2km
Josephiswalking at
h EverytimeJosephwalks1km heearns 5

I Josephwalked
m moneyJosephearnedbywalking
wdependson mdepends on w

males 5kt not

mg
we met lot

dw 2dt dm5dw dm 10 dt

12 few 5 10

If 10 52 fE

mother
example Forfins x2 glow sinx let him g fin g x2 sin x2
Whatis him
If as x changes a littlebithowmuch is hchanging

ne
d
line
f
t line
44
line line ftp.nqcoudu
gf
 dlsinus cosudu.ua du
2xdocdcsincx3
coscx32xdsc

h hin IgE coset 2x fu

CHAINRULE

fog cx feger f'cgesusg.cn

)
sins 42 a(
9 inx x

24 = 2u
dU
Ix
2u
2sinxcosx

sin3x sinu u 3x
3
cosuff
cos u

cos3 3

3cos3x

43 2 1 44 a 43 2 1

443 4 8,4 12 2 2
4143 2 17 12 2 2

Goalfind Ccosx

sincatbs
sinacosbtcosasinbsincxtIS
sinxficosx.si cosse

cossi sincatE U SHE

sinus 1
COsu

cos E
cosxcosE sinxs.in

sink
f sinx u sinx

u1 dy cosse
14 2

t.tt

un u fu
1.4 2 9 III'd
fix

f
f fun.fm
fin ge for gig
fix for 92
t
fi
gIs9
)
QUOTIENTRULE (

gi9
3 432
II
3 1
7 1

Han
r
Ej
os
II

seek
 QUIZ 3

For the following questions, you may use any derivative rules we found in class, unless
stated otherwise in the question.

1. (5 pts) Find f 0 (x) when f (x) = 3x3 2x.

Solution.
f 0 (x) = (3x3 2x)0
= (3x3 )0 (2x)0
= 3(x3 )0 2(x0 )
= 3(3x2 ) 2·1
2
= 9x 2
⇤
2. (5 pts) What is the slope of the tangent line to f (x) = x2 x when x = 2?
Solution. The slope of the tangent line to f (x) at x = 2 is f 0 (2). Since
f 0 (x) = (x2 x)0
= (x2 )0 (x)0
= 2x 1,
the slope of the tangent line to f (x) at x = 2 is
f 0 (2) = 2(2) 1 = 3.
⇤
3. (Extra Credit: 2 pts) What is the equation of the tangent line to f (x) = x2 x when
x = 2?
Solution. The slope of the tangent line is 3, while the tangent line meets the curve
f (x) when x = 2, so the line must go through the point (2, f (2)). We can calculate
that
f (2) = (2)2 2 = 4 2 = 2,
so the equation of the line must be
y = 3(x 2) + 2 = 3x 6 + 2 = 3x 4.
⇤
eviewn

DifferentiationRules DifferentiationRules commonfunctions

patterns

f g 1 nx 1 net
fig
f cf sins cosx

fg f'g fg rule
product Cosa sink

f 92ft quotientrule tana

x)=
seek
secaa
o

RACTICE

1 fix x sinx

2 fix Fccos a
3 for
4 for I
5 fois tan 1 2

answer

1 ofsina E sinat se sinus

2xsinxtsecos.sc

2 secossi E cos sit siccossi

2xcos it x cosu u x
y 2x
2xcossi x sina.ly
2xcos I ofsin i 2x

2xcos I 2 3sinx

Iiiiii
 Ctann'c1tc
4 CI Is.Igxcetcosas
seixctt sin

1
4,95
s.at D c anuiu
seczu.fr
sec
ji
e

hainruleisveryusefulforfindingnew derivativerules

POWERRULE

y fin E find 1 fin y fcn x for a.be

Fc x
1,1 zy Find y f'd

y x y fca x
Laa
eye c ya xb
291 1 yoga d
113
ayatfy bxb1
I.sc g
xbeyea
findfy fin baxbicxkya
y fu
fI x baxbesct.ua
y 113
y x baxb.it cras

Cy3 x basket b
1
35 1 1 tax
It x x if n xn non
Is rulefor
Nowwecanusepower neQrationalnumbers
x

for x fix 3 3 1 3oct

PRACTICE

1 fix
2 fix cost
3 fin
4 few xI

ff.EE j zx31 3x3 3E

2 cost UE u cosx If sink

In
IF Csina

3
CIT LET
2xMAIL.EE u x ty If 32
2 23 EU
7
2 23 I
1
2 23
1 att 3 2
2x 235
4K 34

34
I
I
S
4 1 55 Tu U xtE
11 E 1 15 1t

In fi

2 7