Knf1013 Week9 Differentiation Compatibility Mode

8/2/2019 Knf1013 Week9 Differentiation Compatibility Mode

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KNF1013KNF1013 Engineering MathematicsEngineering Mathematics II

DIFFERENTIATIONDIFFERENTIATION

Dr. Ivy Tan Ai Wei

Ext. 3312

1

. .


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Introduct ionIntroduct ionFirst order derivat ive of f (x)First order derivat ive of f (x) The first order derivative of the function f(x) at x=a isThe first order derivative of the function f(x) at x=a is

hafhafaf

h)()(lim)('

0

+=

provided the limits exists.provided the limits exists.

If the limit exists we sa thatIf the limit exists we sa that ff is differentiable atis differentiable atx = a.x = a.

differentiation. Common notation isdifferentiation. Common notation is

or f(x)or f(x)df

dx

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Example 1: Use the definition to find f (x)2

xhx +

hxf

hlim)('

0=

22

hxf

hlim)('

0=

h

xhxhxxfh

121)2(2lim)('0

+++=

3


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xhxhx 121242'

222 +++

h

xh

m0

=

xhx

h

hxh

h

hxhxf

hhh

424lim)24(

lim24

lim)('000

=+=+

=+

=

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Example:

Use the definition to find f (x)=

3

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Basic Rules of DifferentiationBasic Rules of Differentiation1.1. Constant Mult iplicat ion RuleConstant Mult iplicat ion Rule

==

ddx

dxy

dx==

Example: Given y = 5x. Find the derivative of 2y.Example: Given y = 5x. Find the derivative of 2y.

10)5(22)2( ===dx

dyy

dx

d

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2.2. Sum RuleSum Rule

,,

can be dealt separately.can be dealt separately.

[ ] )()()()( xgdx

dxf

dx

dxgxf

dx

d+=+

[ ] )()()()( xgd

xfd

xgxfd

=

xxx

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Example 1: y = xExample 1: y = x44 + x+ x22

xxdy

243 +=

44 -- 22

d

xxdx =

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3.3. Product RuleProduct Rule

,,

differentiate f(x).g(x)differentiate f(x).g(x)

[ ] ))(()())(()()().( xfdx

dxgxg

dx

dxfxgxf

dx

d+=

oror

'

. xxgxgxxgx +=

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Example 1: y = (4xExample 1: y = (4x22 1)(7x1)(7x33 + x)+ x)

)8)(7()121)(14(322

xxxxxdx

dy

+++=

= 84x4 + 4x2 21x2 -1 + 56x4 + 8x2

24 =

and differentiate in normal way.and differentiate in normal way.

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Example 2: If f(x) = xExample 2: If f(x) = x22 tan x. Find f (x)tan x. Find f (x)

Using product ruleUsing product rule

f (x) = xf (x) = x22

[sec[sec22

x] + tan x (2x)x] + tan x (2x)

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Find t he der ivat ive of

2) (3 2 )(5 4 )a f x x x x= +

2

) ( ) 3 sinb f x x x=

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4.4. Quot ient RuleQuot ient Rule

If f x and x are differentiable at x for and x 0.If f x and x are differentiable at x for and x 0.)(xf

ThenThen)(xg

2

))(()())(()()(

x

xgdx

xfxfdx

xg

x

xf

dx

d

=

))(')()(')( xgxfxfxg =

)]([ xg

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2

Example 1:Example 1: . Find1

2 +=

xy

dx

22

2222

)1()2()2()1( ++=xdx

dxxdx

dxdy

22

)2)(2()2)(1( += xxxxdy)1( +xdx

334222 ++ xxxxd

22)1( += xdx

22

)1(+

=

xdx 14


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Examples:Examples:

i) Show that2(tan( )) sec ( )d x x

d x=

i i ) Different iat e

s iny

x=

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1

)(

=

nn

nxxdx

Example 1: Find the derivative of f(x) = xExample 1: Find the derivative of f(x) = x33

f x = = 3xf x = = 3x22)(3x

d

x

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Example 2:Example 2:

== 88 == 107107

Solution: We haveSolution: We have

f(x)=f(x)=7188

88 xxxd

==

SimilarlySimilarly

g(t)=g(t)=1061107107

107107 tttdt

d==

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Ex m l :Ex m l : If y = 2xIf y = 2x44 2x2x33 xx22 + 3x+ 3x 2, find2, find dx

dy

)2()(3)()(2)(2 234dxdyx

dxdx

dxdx

dxdx

dxd

dxdy +=

0)1(32)3(2)4(223 += xxx

326823 += xxx

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Chain RuleChain Rule

If y = f (u) and u = g(x), then y = f (g(x)) and the chainIf y = f (u) and u = g(x), then y = f (g(x)) and the chainrule says thatrule says that

dudydy.= xux

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Differentiate y = (xDifferentiate y = (x33 + x+ x 1)1)55..

For u = x3 + x -1, note that y = u5.

13 2 += xdxdu 45u

dudy =

dx

du

du

dy

dx

dy.=

)13(524 += xu

dx

dy

)13()1(5243 ++= xxx

dy

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Example 2: FindExample 2: Find if y = 4if y = 4 coscos xx33dx

dy

Let u = xLet u = x33

y = 4y = 4 coscos uu2du dy

dx du

dudd

dxdudx.=

)3)(sin4(2

xudx =

32sin12 xx

dx

=

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Examples: Find the derivative ofExamples: Find the derivative of

2

cosa y x=

=2

) cos(3 )c y x=2

) cosd y x=

) cose y x=

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ReviewReview

Di erentiateDi erentiate =4

)5

a y

x

= 43 tan 3y xb)b)

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Differentiation of Circular FunctionsDifferentiation of Circular Functions

= .

From the defini t ion, w e find t hat

hxf

hlim)('

0

=

hh

lim0

=

hhlim0=

24


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Grouping terms w ith sin(x ) and terms w ith sin(h)

separately

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

x h x x h= +

0 0h hh h

cos 1 sinh h0 0

s n m cos mh h

x xh h

= +

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Thus, w e find t hat t he derivat ive of f( x) = sin (x)

sin (0) cos (1)x x= +

We w rit e,

cos x=

)(cos))(sin( xxxd

d=

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The derivatives of all six trigonometric functions areThe derivatives of all six trigonometric functions are

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Show thatShow that [cos ] sind

x x=x

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Differentiation of Inverse FunctionsDifferentiation of Inverse Functions

Let us consider now finding the derivatives of f(x) = sinLet us consider now finding the derivatives of f(x) = sin--11

x .x .

Let y = sin-1

(x)

)]([sin xdxdx

=

nstea o erent at ng y = s nnstea o erent at ng y = s n-- x , we erent ate x =x , we erent ate x =sin y with respect to x. We obtainsin y with respect to x. We obtain

=

][sin)( ydx

dx

dx

d=

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Applying the chain rule to the right hand side, we find thatApplying the chain rule to the right hand side, we find that

Now, sinNow, sin22(y) + cos(y) + cos22(y) = 1(y) = 1dx

y][cos1=

coscos22(y) = 1(y) = 1 sinsin22(y)(y)

coscos (y) =(y) = since x = sin ysince x = sin y)(sin12

y

We also know that yWe also know that y

2,

2

hencehence coscos (y) , Thus, we cannot take y=(y) , Thus, we cannot take y= -02

1 x

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we takewe take coscos (y) =(y) = 21 x

It follows that,It follows that,dy

dx

dy2

dxx=

1dy

21 xdx

=

2

1

1

)(sin

x

x

dx

=

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The derivatives of all six inverse trigonometric function areThe derivatives of all six inverse trigonometric function are

1.1. , for, for --1 < x < 11 < x < 11 1

)(sin xd

=

2.2. , for, for --1 < x < 11 < x < 1

x

2

1 1

)(cos xdx

d

=

3.3.2

1

1

1)(tan

xx

dx

d

+=

4.4.2

1

1

1)(cot

xx

dx

d

+

=

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5.5. , for |x| > 1, for |x| > 11||

1)(sec

2

1

=

xxx

dx

d

6.6. , for |x| > 1, for |x| > 111 =d

1||2 xxdx

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Exam le 1:Exam le 1:

Find for y = sinFind for y = sin--11 3x ( using the chain rule)3x ( using the chain rule)dx

dy

y = sin-1 3x

)]3([)3(1

2x

dxxdx

y

=

]3[)9(1

1

2x=

3=

)9(12x

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Example 2:Example 2:

Find for y = cosFind for y = cos--11(2x(2x22 + 1)+ 1)1010dx

dy

y = cos-1

(2x2

+ 1)10

1 dd

)12(1202

++

= xdxxdx

)]12(.)12(10[)12(1

1 292202

+++

= x

dxx

x

)]4.()12(10[1 92

202xx +

=

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Compute the derivative of :Compute the derivative of :

1 2) cos (3 )a x

1 3

)(sec )b x

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Knf1013 Week9 Differentiation Compatibility Mode