3.1 Derivatives of Polynomials andExponential Functions

Polynomials

By considering derivatives for 113 34

let's see if we can find a pattern forLet f x X Hxth Cath 2 x72xhth2

Then f't highHxthh

Lingo Ht2xhhth XT

thief 2xhh

Lingo h 2xyththinfo 2x 2 0 2

Let f x _x3 faith CxthP tht

In't t thThen f x thiefHxthh

thief 3 4 34743 13X3t3xht3xhth3

thief 3 4 3 42 thh

thief1h13x'thyxhthLingo 13 73 4 3 2

Let f x x 4 c Hxth Hh If h xp

then f x lining Hxth Hx X t4xh TIh

high t4x3ht6xh7h4xh th 14KLingo 4x3ht6x2h74xh3i

hthing till4 3 6 2ht4xh2th3

HLingo 4 76 44 1784 3

DOWe see that Ex 2 x

dddefirative dqEx 3

2

withrespectto X f

X 4 3

we can see the patternpowerRule

Cx nxn 1

To prove this use the Binomial Theorem

Examples find the derivative ofeach of the following

Hx Ff x 5 5

1 5 4

xI x 1 0 1 I 1

Derivatives glx 7 9 3 4distribute

IatIitisofbtraction9 7695,1132443

or Ix's

y 2 40 8 12

4 y 2.40 39 8.12 x80 39 96 X

what about the derivative of a

constant fC horizontal line

my faith c

s c 5C f G LieffCxthhi X lim

Sincetheslope h70 hSfpd theFahnegentline nF'of

e Lingo0 0

Yet another way to think about thederivative of a constant63 Cc ca.no i IoEo

More examples Rewrite first

Hx IT 3 3

f x l xt t

OXZ

12

F2 12 7512 521 3zx t 7 tx

tt 0

3 2 72

32

gig 23 fryb Fbm 2 Forbnto y zy.is

Algehw

iz 3Fcalculus g ly f Ey

tz y

t I

Igy Et yAlgebra

psstI

ly'si ftp.t offs'sgo.y.IT I 1 or

q3g'RE yty g y

fix 3xstfu fpsftf.si 54

3 5 2 1 5 4 128 S

f x 3 5 4 2 1 2 5 244 t's 8

80

15 4 2 2

thy X E 4 9

415 4 45 TaTTxt 41x4fx3 3

415 4ty Ti

y K312 3

bmqbm

x 3

1 AlgebraYi x'T Y 32 4

E X E 3 2

y g72x 3 2

2

calculus

IzI 6 3

27 1 t 3

Trxa x4rx Algebra

rationalize r T727fi7Tx 12 2

2 5 t 275

75 12 2

2 5

DEI e is the number suchthat high ehhI 1

know e k 2.7182

we can use this definition toget the derivative of exLet fix exThen f x Inigo Hxth f

hmm Inigo exthexbm.brb h

Lingo e eh_e

ah

Info

ex Lingo E

ex I

ex

So now we know g e ex

Examples find the derivativeHx ex Sxf X Ex GS

x 5 0 5

L 3e t I

dot X T 3 e t I

IFS 3 text O

IzXi's 3ex p

3ex

At what pointG on the curve

y x3t3 2is the tangent line

horizontalderivative

slope O

y x't 3 2

y 3 2 t 3 2X 3 2 6X O3 1 2 O

3X O Xt 2 03 70I

25 3125y 03302 0g 12 4

0,0 2,4

find the equation of thetangent line to the curve y x rxat the point 1

isfind slope

y y mix m y'lly O m x 1 y X Xk

M y l Ext I

Y try x l L Ex

y tax ty Trxy'll I It

Find the equation of the tangentline to the curve g 2e tx3 atthe point 10,2 Find slopeX y

Y y m xY'co

Y 2 ne x o Y 2e't 3 2

y 2 2782

2 t 2 ORdye 2e t3x

y 2 2

y 2Et3