Chapter 2 – Part I - The Derivative

CHAPTER 2 – CHAPTER 2 – PART IPART I

The DerivativeThe DerivativeFaculty of Science & TechnologyFaculty of Science & Technology

pages : pages :

KG/ME_SST/SEPT08KG/ME_SST/SEPT08 22

OUTLINEOUTLINEThe Tangent LineThe Tangent LineDefinition of The DerivativeDefinition of The DerivativeRules for Finding DerivativesRules for Finding DerivativesDerivatives of Trigonometric FunctionsDerivatives of Trigonometric FunctionsThe Chain RuleThe Chain RuleHigher-Order DerivativesHigher-Order DerivativesImplicit DifferentiationImplicit DifferentiationApplications of The DerivativeApplications of The Derivative


Tangent DefinitionTangent DefinitionFrom geometryFrom geometry– a line in the plane of a circlea line in the plane of a circle– intersects in exactly intersects in exactly oneone point point

We wish to enlarge on the idea to include We wish to enlarge on the idea to include tangency to any function, f(x)tangency to any function, f(x)


Slope of Line Tangent to a Slope of Line Tangent to a CurveCurve

Approximated by secantsApproximated by secants– twotwo points of intersection points of intersection

Let second point get closer and closer to Let second point get closer and closer to desired point of tangencydesired point of tangency

•• •


Animated Secant LineAnimated Secant Line


Animated TangentAnimated Tangent


Slope of Line Tangent to a Slope of Line Tangent to a CurveCurve

Recall the concept of a limit fromRecall the concept of a limit from previous chapter previous chapterUse the limit in this contextUse the limit in this context ••

0 0

0

( ) ( )limx

f x x f xmx

x


The Slope Is a LimitThe Slope Is a LimitConsider f(x) = xConsider f(x) = x33.. Find the tangent at Find the tangent at xx00= 2= 2

Now finish …Now finish …

0

3 3

0

2 3

0

(2 ) (2)lim

(2 ) 2lim

8 12 6( ) ( ) 8lim

x

x

x

f x fmx

xmxx x xm

x


Definition of DerivativeDefinition of DerivativeThe derivative is the The derivative is the formulaformula which which gives the slope of the tangent line at gives the slope of the tangent line at any point any point xx for for ff((xx))

Note: the limit Note: the limit must existmust exist no holeno hole no jumpno jump no poleno pole no sharp cornerno sharp corner

0 0

0

( ) ( )'( ) limh

f x h f xf x

h

A derivative is a limit !


Derivative NotationDerivative NotationFor the function For the function yy = = ff((xx))Derivative may be expressed as …Derivative may be expressed as …

'( ) " prime of "

"the derivative of with respect to "

f x f xdy y xdx


FindFind f f ’’((xx) ) using definition.using definition.

0

0

0

0 0

[sin( ) 1] [sin 1]'( ) lim

sin( )cos( ) cos( )sin( ) 1 sin 1lim

sin( )(cos( ) 1) cos( )sin( )lim

sin( ) cos( )

cos(

(cos( ) 1) sin( )lim

( ) s

i

)

l

1

m

in

h

h h

h

h

x h xf xh

x h x h xh

x h x h

f x x

h hh

h

x x

xh


Differentiability implies Differentiability implies continuity.continuity.

If the graph of a function has a tangent at If the graph of a function has a tangent at point c, then there is no “jump” on the point c, then there is no “jump” on the graph at that point, thus is continuous graph at that point, thus is continuous there.there.

If ' , then is continuous at f c f c


A Derivative is a LimitA Derivative is a LimitTherefore, the rules for limits, Therefore, the rules for limits, essentially become the rules for essentially become the rules for derivatives.derivatives.Derivative of a sum/difference is the Derivative of a sum/difference is the sum/difference of the derivatives.sum/difference of the derivatives.Derivative of a product/quotient is the Derivative of a product/quotient is the product/quotient of the derivatives.product/quotient of the derivatives.


Basic DerivativesBasic DerivativesConstant FunctionConstant Function– Given f(x) = kGiven f(x) = k– Then f’(x) = 0Then f’(x) = 0

Power FunctionPower Function– Given f(x) = x Given f(x) = x nn

– Then Then 1'( ) nf x n x

( ) 0d kdx

1n nd x n xdx


Basic DerivativesBasic Derivatives

Identity FunctionIdentity Function– Given f(x) = xGiven f(x) = x– Then f’(x) = 1Then f’(x) = 1

( ) 1d xdx


Basic RulesBasic RulesConstant multipleConstant multipleRuleRule

Sum RuleSum Rule

DifferenceDifferenceRuleRule

( ) ( )d dc f x c f xdx dx

( ) ( ) ( ) ( )d d df x g x f x g xdx dx dx

( ) ( ) ( ) ( )d d df x g x f x g xdx dx dx


Try It OutTry It OutDetermine the followingDetermine the following

2 2 3' 2 2

y t ty t

( ) 3 5

'( ) 3

f x x

f x

3

4

( )

'( ) 3

p x x

p x x

3

4

143

( )

11 '( )3

xh xx

h x x


Product RuleProduct RuleConsider the product of two functionsConsider the product of two functions

It can be shown thatIt can be shown that

In words:In words:– The first function times the derivative of the second The first function times the derivative of the second

plus the second function times the derivative of the plus the second function times the derivative of the firstfirst

( ) ( ) ( )f x h x k x

'( ) ( ) '( ) ( ) '( )f x h x k x k x h x


Quotient RuleQuotient RuleWhen our function is the quotient of two other When our function is the quotient of two other functions …functions …

The quotient rule specifies the derivativeThe quotient rule specifies the derivative

In words:In words:– The denominator times the derivative of the numerator The denominator times the derivative of the numerator

minus the numerator times the derivative of the minus the numerator times the derivative of the denominator, all divided by the square of the denominatordenominator, all divided by the square of the denominator

( )( )( )

p xf xq x

2

( ) '( ) ( ) '( )'( )( )

q x p x p x q xf xq x


Just Checking . . .Just Checking . . .Find the derivatives of the given functionsFind the derivatives of the given functions

2

3 2

3

2

2 3 6 3

' 2 3 6 6 3 3p

p x x x

x x x x x

2

2

22

5 7 7 4 2 ( )

5

7 4( )5

xq x

xq x

x

xx x


Derivatives of Trigonometric Derivatives of Trigonometric FunctionsFunctions


Find the derivative of the Find the derivative of the followingfollowing

2

2

2

2

(sec tan csc ) (sec cot )(1)'

sec tan csc sec cot'

sec cot

x x x x x xyx

x x x x x x xyx

x xyx

The Chain RuleThe Chain Rule


Solution: The Chain RuleSolution: The Chain RuleGiven y = f (u) and u = g (x)Given y = f (u) and u = g (x)– That is y = f(u) = f ( g(x) )That is y = f(u) = f ( g(x) )

ThenThen

In words:In words:– The derivative of The derivative of y with respect to xy with respect to x is is

the derivative of the derivative of y with respect to uy with respect to u timestimesthe derivative of the derivative of u with respect to xu with respect to x

dy dy dudx du dx


Chain RuleChain RuleExampleExample – given – given– Then andThen and

34 28 3y x x 3( )y f u u 4 28 3u x x

dy dy dudx du dx

34 2 3'( ) 3 8 3 32 6f x x x x x


Other ExampleOther ExampleGiven:Given:y = (6xy = (6x33 – 4x + 7) – 4x + 7)33

Then u(x) = 6xThen u(x) = 6x33 – 4x + 7 – 4x + 7and f(u) = uand f(u) = u33

ThusThusf’(x) = 3(6xf’(x) = 3(6x33 – 4x + 7) – 4x + 7)22(18x(18x22 – 4) – 4)

dydu

dudx


Find the derivativeFind the derivativeNote this is the composition of 3 functions, Note this is the composition of 3 functions, therefore there will be 3 “pieces” to the chain.therefore there will be 3 “pieces” to the chain.

3

3 5

3 4

2

5

' cos[( csc ) ]

[5( csc ) ]

(3 csc c

sin[( csc

)

) ]

ot

y x x

x x

x x

y x

x

x


Derivatives of Exponential Derivatives of Exponential FunctionsFunctions

ConclusionConclusion

When y = aWhen y = ag(x)g(x)

– Use chain ruleUse chain rule

Similarly for y = eSimilarly for y = eg(x)g(x)

ln( )x xxD a a a

( )ln '( )g x

dy dy dudx du dx

a a g x

( ) ( ) '( )g x g xxD e e g x


Derivative of the Log Derivative of the Log FunctionFunction

For the natural logarithm For the natural logarithm ln(x)ln(x)

–

For the log of a different base logFor the log of a different base logaa(x)(x)

–

1lnxD xx

1log

lnx aD xa x


What About ln(-x) ?What About ln(-x) ?Consider it a compound functionConsider it a compound function

Apply the chain ruleApply the chain rule

Thus we see Thus we see

( ) ln( ) ( )( ( ))

f x x g x xy f g x

1 ( ) 1 1ln( ) 1xd xD x

x dx x x

ln( ) ln( )x xD x D x


Differentiate each of the Differentiate each of the following functions.following functions.


Higher-Order DerivativesHigher-Order Derivativesf’’ = 2f’’ = 2ndnd derivative derivativef’’’ = 3f’’’ = 3rdrd derivative derivativef’’’’ = 4f’’’’ = 4thth derivative, etc… derivative, etc…

The 2The 2ndnd derivative is the derivative of derivative is the derivative of the 1the 1stst derivative. derivative.

The 3The 3rdrd derivative is the derivative of derivative is the derivative of the 2the 2ndnd derivative, etc… derivative, etc…


Alternate NotationAlternate NotationThere is some alternate notation for higher There is some alternate notation for higher order derivatives as well. order derivatives as well. Recall that there was a fractional notation Recall that there was a fractional notation for the first derivative.for the first derivative.

We can extend this to higher order We can extend this to higher order derivatives :derivatives :

… etc.


Example :Example :Find the first four derivatives for each of Find the first four derivatives for each of the following.the following.


Solutions :Solutions :(a)(a) (b) (b)


Implicit DifferentiationImplicit DifferentiationConsider an equation involving Consider an equation involving bothboth x and y: x and y:

This equation This equation implicitlyimplicitly defines a function in x defines a function in xIt could be defined It could be defined exexplicitlyplicitly

2 2 49x y

2 49 ( 7)y x where x


DifferentiateDifferentiateDifferentiate both sides of the equationDifferentiate both sides of the equation– each termeach term– one at a timeone at a time– use the chain rule for terms containing yuse the chain rule for terms containing y

For we getFor we get

Now solve for dy/dxNow solve for dy/dx

2 2 49x y

2 2 0dyx ydx

dy xdx y


Differentiate cont.Differentiate cont.Then gives usThen gives us

We can replace the y in the results with the We can replace the y in the results with the explicit value of y as neededexplicit value of y as neededThis gives usThis gives usthe slope on the the slope on the curve for any curve for any legal value of xlegal value of x

2 2 0dyx ydx

22

dy x xdx y y

2 49

dy xdx x


Second DerivativeSecond DerivativeGiven xGiven x2 2 –y–y22 = 49 = 49

y’ =??y’ =??

y’’ =y’’ =

' xyy

2

2 2

'd y y x ydx y

Substitute


Find the derivativeFind the derivative

2

2

2

2

cos( ) ( 1) sec ( ) 2

cos( ) cos( ) sec ( ) 2

( cos( ) sec ) 2 cos( )

2 cos( )cos( ) s

sin( ) tan )

ec

( 2dy dyxy x y ydx dx

dy dyxy x y xy ydx dx

dy x xy y y xydxdy y xydx

xy y

xy

x

x y

Applications OfApplications OfThe DerivativeThe Derivative

Refer pages 151-214 : Chapter 3

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Chapter 2 – Part I - The Derivative

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