higher derivative

7/29/2019 Higher Derivative

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Calculus 3rdsecondary H igher Deri vative Part II -7-

n n 1

:

From our previous study , we have known that : If y is a differentiable function with respect to x

Then the derivative of :

dyy y' n y

dx

Introduction

Simple3 2d: y 3y " which is wrong " W hy?!!!!

dx

Because when we differentiate and the answer doesn't include xdy

Then we must add in order to respect that xdx

example

wi th respect to x

3 2

3 2

d dySo y 3y Thisis the right answer

dx dx

dBut y 3y Only Thisis the right answer

dy

d dy dy1 x y x y 1 x ydx dx dx

2

Examples on der ivatives wi th respect to x

2 3 2 2 3 2 2 3

2

6 5 32 2 2

6 61 -1 1 17

2 2 2 2

d dy dy2x y 2x 3y y 4x 6 x y 4x y

dx dx dxdy

y xd x dx3dx y y

d4 x 3x 2 6 x 3x 2 2x 3 12x 18 x 3x 2

dx

d 1 dy x dy5 1 2x y 7 1 2x y 2x y y 2 7 1 2x y 2 y

dx 2 dx dxy

----------------------------------------------------------------------------------------- --------------

mplicit Differentiation


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Calculus 3rdsecondary H igher Deri vative Part II --

Example (1)

dyFind in each of the following

dx

2 2a x 4 x y 3y 2x 5 Answer

Differentiate with respect to x :

dy dy dy dy2x 4 x 4y 6 y 2 0 " Divideby2" x 2x 2y 3y 1 0

dx dx dx dx

dy dy 1 2y x2x 3y 1 2y x

dx dx 2x 3y

3 2 2b y 2x y 3x y 0 Answer

2 2 2 2 2 2

22 2 2

2 2


dy dy dy dy dy dy3y 2x 4x y 3x 2y 3y 0 3y 2x 4x y 6 x y 3y 0

dx dx dx dx dx dx

dy dy 3y 4 x y3y 2x 6x y 3y 4 x y

dx dx 3y 2x 6x y

c Sin3 y Cos2x 0 Answer


Sin3 y Cos2x 0

dy dy dy -2Sin2x3 Cos3y 2Sin2x 0 3 Cos3y -2Sin2x

dx dx dx 3Cos3y

d xCos y ySin2x 1

Answer


dy dyx -Sin y Cos y 2yCos2x Sin2x 0

dx dx

dy dy- xSin y Cos y 2yCos2x Sin2x 0

dx dx

dy dy -Cos y 2yCos2xSin2x xSin y -Cos y 2yCos2x

dx dx Sin2x xSin y


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4 3 2

23 2 2

2

3 4 5

3 4 5

1If y x 2x x 3x 1

4

dy d yThen : y' x 6x 2x 3 y'' 3x 12x 2

dx dx

d y d y d yAnd : y''' 6x 12 y'''' 6 y'''' 0

dx dx dx

--------------------------------------------------------------------------- ----------------------------

Example (2)2

5

2

d yIf y Z And Z 2x 3 , Then find

dx

Answer

4 4 4

24 3 3

2

dy dz dy dy dz 5Z And 2 5Z 2 10Z

dz dx dx dz dx

dy d yThen 10 2x 3 40 2x 3 2 80 2x 3

dx dx

----------------------------------------------------------------------------------------- --------------Example (3)2If 6x 5 4x y , Then Prove that : x y'' 2y' 3

Answer

Differentiate with respect to x : 12x 4x y' 4y " by 4" 3x x y' y

Differentiate again with respect to x : 3 x y'' y' y' x y'' 2y' 3

----------------------------------------------------------------------------------------- --------------Example (4)

24 3

2

2x

If 3y 4x , Then Prove that : y y'' 3 y' y

Answer

3 2 3 2

23 2 3 2

2 22

2

Differentiate with respect to x : 12y y' 12x y y' x

Differentiate again with respect to x : y y'' y' 3y y' 2x y y'' 3y y' 2x

2xy y y'' 3 y' 2x y y'' 3 y'

y

----------------------------------------------------------------------------------------- --------------

Higher Derivatives


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Example (5)

If y xCos x Sinx , Then Prove that : x y''' x y' 2y 2Sinx

Answer

y' -xSinx Cosx Cosx -xSinx 2CosxAnd y'' -xCosx Sinx 2Sin x - xCos x Sinx 2Sin x

And y xCos x Sinx y'' - y 2Sin x

y'' - y 2Sin x " Multiply both sides by x " x y'' - x y 2xSinx

So by differentiating again with respec

t to x :

x y''' y'' - x y' y 2 xCosx Sinx x y''' y'' - x y' y 2y

x y''' y 2Sin x - x y' y y x y''' x y' 2y 2Sinx

------------------------------------------------------------------------------- ------------------------Example (6)

xFind y''' if y , where x 1

x 1

Answer

-2 -3 -4

2 2 4

x 1 x -1 -6y' - x 1 y'' 2 x 1 y''' -6 x 1

x 1 x 1 x 1

----------------------------------------------------------------------------------------- --------------Example (7)

Answer

2

2 2 22 2

2 2 2

Differentiate with respect to x

- x2x 2y y' 0 " by 2" x y y' 0 y' 1

y

- y x y'

Differentiate again with respect to x : y'' 2y

So by substituting 1 in 2 :

- x x y x 1- y x - y - - y x

y y y y yy''

y y y y

2 3

-25

y

----------------------------------------------------------------------------------------- --------------

2 2

3

-25If x y 25 , Prove that y''

y


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Example (8)2

3 4

2

d zIf y 8x 1 And Z 6x , then find at x 1.5

dy

Answer2

2

3 2 4 3

3

2

2

2

d zmeans that we have to differentiate with respect to y

dy

dy dz So for y 8x 1 24x And for y 6x 24x

dx dx

dz dz dx 24xAnd x Then differentiate again with respect to y

dy dx dy 24x

d z dx

Then dy

2

22 2

1 d z 1 1

So at x 1.5dy 24x dy 5424 1.5

----------------------------------------------------------------------- --------------------------------Example (9)

22

2

dz dy d z If 3 2x And x 3 , Then find At x 2

dx dx dy

Answer

22

2

2

2 2 2

2 2 2

2 2 32 2 2 2

2 2

2 2

dz dy d z If 3 2x And x 3 , Then find At x 2

dx dx dydz dz dx 3 2x

So Differentiate with respect to y :dy dx dy x 3

dx dx dx2 x 3 2x 3 2x 2x 6 6x 4x

d z 2x 6 6x 4xdy dy dy

dy x 3 x 3 x 3

d z -2x 6 6x

dyx

22

3 32 2

-2 2 6 6 2d z -26 So at x 2 -26

dy 13 2 3

----------------------------------------------------------------------------------------- --------------


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Example (10)

3

73

2

dx x 2If y x 2 , Then Prove that :

dy 21x y

Answer

6 63 2 2 3

3 3

6 7 22 3 2 3


dy7 x 2 3x 21x x 2

dx

x 2dx 1 x 2

dy 21x y21x x 2 21x x 2

--------------------------------------------------------------------------- ----------------------------

Example (11)

2

32 2

2

d yIf y x 1 x 2 And Z x 7 x 2 , Then prove that : 2x 5 6x 30x 22

dz

Answer

2 2 2 2

2

2

22

22

22

2

dyx 1 1 x 2 2x x 1 2x 4x 3x 4x 1

dx

dzx 7 1 x 2 1 x 7 x 2 2x 5

dx

3x 4x 1dy dy dz dz dx dx 2x 5

So differentiate again with respect to Z :

dx dx2x 5 6 x 4 3x 4x 1 2

d y dz dz

dz 2x 5

dx2x 5 6x 4 2 3x 4x

d y dz

dz

2

2 2 2 2

2 22

23 2

2

1

2x 5

d y 12x 8x 30x 20 6x 8x 2 6x 30x 22

dz 2x 5 2x 5 2x 5 2x 5

d y2x 5 6x 30x 22

dz

----------------------------------------------------------------------------------------- --------------


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Example (12)

2

22

2x 1 x dy -2 dz If y And Z , Then Prove that :

x dx Z dxx 1

Answer

12 2

12 2

1 1-

2 2 1 12 2 -2 2 22 2

2 212 2

12 2 22

322 2

1 1- 12 22 2 -

2 2 22

2

x 12xy , Z

xx 1

1x 1 2 2x x 1 2x

2 x 1 2x x 12dy

dx x 1

x 1

2 x 1 x 1 xdy 2

dx x 1x 1

1x x 1 2x x 1

x x 1 x 1dz 2

dx x

11-

2 3 222

2 2

12 2 2

21

2 2 2 2

2 2 2 2

2

122 2 2

x 1 x x 1

x x

dz -11

dxx x 1

x 1-2 x 1 -2 -2 x-2 -2 2

Z x x Z x 1

dz -2 -1 -2 x

Multiply 1 and 2 : dx Z xx x 1

322 2

2

1x 1

----------------------------------------------------------------------------------------- --------------


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12 2 2

-12 2

1 2 22 2

2

2

12

2 2

2

x 1

xAssume y 8 x 8 x And Z

x 1

x 1 xdy 1 x dz 18 x 2x And

dx 2 dx x 1 x 18 x

x x 1dy dy dx xx 1

dz dx dz 8 x8 x

1 1 1dy 4So at x 1:

dz 38 1

2 x

Find the rate of change of 8 x with respect to at x 1x 1

Example (13)

23

2 2

2 2 2 2

xif f x y And Z , Then Prove that:

y

1 dz 1 dy 2 d z dy dz d y -2i ii y 2 Z

Z dx y dx 3x dx dx dx dx 9y z

Answer

-1 2 -1 -12 2 2

3 3 3 33 3 3

2 2 2 2

-12

33

2 2 2

3 3

2 dy 2 2x y x x y x

x dz dz x dy x dy3 dx 3 3i Zy dx y dx y y dx y y dx

2x

y1 y 1 dz y x dy3AndZ Z dx y y dx

x x

-1

3

2

3

2x

3y

x

y 23x

2

3

x 2y

22 -133 3

2

2

dydx

1 dz 2 1 dy 1 dz 1 dy 2 1 dy 1 dy 2

Z dx 3x y dx Z dx y dx 3x y dx y dx 3x

x dy dz 2ii Z Z y x ,soby differentiate with respect to x : Z y x

y dx dx 3

d y dy dz Differentiate Again with respect to x : Zdx dx dx

-42

32

-2 -22 2

3 322 2 2 22

3

d z dz dy -2y xdx dx dx 9

d z dy dz d y -2 2 -2y 2 Z x x

dx dx dx dx 9 9 y z 9 x

----------------------------------------------------------------------------------------- --------------Example (14)

Answer


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Example (15)

2

32

2 2 d y -8x xIf y x , Z x , Then Prove that :

dzx x x 2

Answer

1 1 1 -3 -3- -

2 2 2 2 2

1 1 1 -3 -3- -

2 2 2 2 2

-3

22

-3 22

2

32 2

2 -3 222

2

dy 1 1y x 2x x x x x 2

dx 2 2

dz 1 1Z x 2x x x x x 2

dx 2 2

dx dx1x 2 x 2x x 2

dy dy dx x 2 d y dz dz 2dz dx dz x 2 dz 1 x 2

x x 22dx

x 2 x 2d y -4 -8 x -8dzdz 1x 2 x 2 x 2

x x 2 x 22

3

x x

x 2

----------------------------------------------------------------------------------------- --------------Example (16)2 22

2

d y dyIf y Cos2x , Then Prove that : 4 16

dx dx

Answer

2

2

22

2 2

2

22 2 2

2 2 2 2

dy d yIf -2Sin2 x -4Cos2x

dx dx

d yL.H.S. : -4Cos2x 16Cos 2x 1

dx

dy4 4 -2Sin2x 4 4Sin 2x 16 Sin 2x 2

dx

from 1 And 2 and by addition :

16Cos 2x 16 Sin 2x 16 Cos 2x Sin 2x 16 R.H.S

----------------------------------------------------------------------------------------- --------------


10/14





Example (17)2

2

d y dyIf x y SinxCos x , Prove that , x 2 4 x y 0

dx dx

Answer

2

2

2 2 2

2 2 2

1 dyx y Sin2x SoDifferentiate with respect to x : x y Cos2x

2 dx

d y dy dyDifferentiate again with respect to x : x -2Sin2x

dx dx dx

d y dy d y dy d y dyx 2 -4SinxCos x x 2 4Sin xCos x 0 x 2 4x y 0

dx dx dx dx dx dx

------------------------------------------------------------------------------------------ -------------

Example (18)2

3 2

2

x 1

d yIf y a x b x And y''' =12 And 6 , Then find a and b

dx

Answer

2y' 3a x 2b x y'' 6 a x 2b 1 y''' 6a

And y''' 12 6a 12 a 2

From 1 : y'' 6 a x 2b Then At x 1 And a 2

6 2 1 2b 6 2b -6 b -3

----------------------------------------------------------------------------------------- --------------Example (19)

22

2

2

dy d y dyIf 2 x y y 3 , Prove that , 2 x y 0

dx d x dx

Answer

2 2

2 2

dy dyDifferentiate with respect to x : 2x 2y 2y 0 2

dx dx

dy dyx y y 0dx dx

d y dy dy d y dy dyDifferentiate again with respect to x : x y 0

dx dx dx dx dx dx

22

2

d y dy dyx y 2

dx dx dx

----------------------------------------------------------------------------------------- --------------


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Example (20)2 2

2 2

d y d x dy dxIf x y a where a is a real positive number and

dx dy dx dy

Then find the interval for which belongs

a

Answer

21 2 3

2

21 2 3

2

-3 -23 3 2 2 2 2

-2

Important note : If you differentiate directly with respect to x , you will not go any where

dy d yy a x -a x 2ax

dx dx

dx d xx a y -a y 2a y

dy dy

2ax 2a y -a x -a y 4a x y a x y

x y4

x y

-3 x y 4 a 4 a 0 ,4

----------------------------------------------------------- ------------------------------ --------------Example (21)

22 2

2

d yIf y x 3x 2 , Z 3x 5x 4 , Then find the value of at x 2

d z

Answer

22 2

2

2

22

2

2

d yIf y x 3x 2 , Z 3x 5x+4 , Then find the value of at x 2

d z

dy dz dy dy dx 2x 32x 3 & 6 x 5

dx dx dz dx dz 6 x 5dx dx

2 6 x 5 6 2x 3d y dz dz Differentiate with respect to Z :dz 6 x 5

dx

12x 1d y dz

dz

2 3

2

2

0 12x 18 -28 6 x 5 6 x 5

d y -28 -4So At x 2

dz 343 49

----------------------------------------------------------------------------------------- --------------


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Example (22)

2

23

1ax x , x 2

2If the function f x is differential twice at x 2x xbx

c , x 2x 3

Then find the value of a , b and c .

Answer

2 2

2 332

2

2

1 1ax x , x 2 ax x , x 2

2 2f x f x

x x xbx

bx c , x 2c , x 2 3x 3

a x , x 2f ' x

2bx x , x 2

And f x is differentiable at x 2

f ' 2 f ' 2 2b 2 2 a 2 4b a 2 1

-1 , x 2f '' x

2b

2x , x 2

And f x is differentiable twice at x 2

3f '' 2 f '' 2 2b 2 2 -1 2b 3 b

2

3Then substitute in 1 : 4 a 2 a 4

2

And f x is differentiable at x 2 , Then it is continous at x 2

f 2 f 2

2 22 2

x 2

3322

x 2

f 2

1 1 1 1f 2 ax x 4 2 2 6 And f 2 lim ax x 4 2 2 6

2 2 2 2

2x 10f 2 lim bx c 1.5 2 c c

3 3 3

10 8Then c 6 c

3 3


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Example (23)

22

xIf f x , Prove that : x 1 f '' x 3x f ' x 0

x 1

Answer

1 1-

2 22 2

1 22 2

11 1-- 2 2 22 2 2 22 2

32 22 2

112 2

2 2

3 3 52 2

2 2

2

2

52 22

1x 1 x x 1 2x

2xf x f ' x

x 1x 1

x 1 x 1 xx 1 x x 1 1f ' x

x 1 x 1x 1

3- x 1 2x

-3x x 1 -3x2f '' x

x 1 x 1 x 1

-3x x 1 3xx 1 f '' x 3x f ' x

x 1 x

3 3 3

2 22 2 2

-3x 3x0 R.H.S

1 x 1 x 1

----------------------------------------------------------------------------------------- --------------Example (24)

dyIf Sin y Cos 2x Tan4x 0 , Then find at x

dx 4

Answer

2

2

2

o o

Sin4x d 4Tan4x Then Tan4x 4Sec 4x

Cos4x dx Cos 4x

dySo , by differentiate with respect to x : Cos y 2Sin2x 4Sec 4x 0 1

dx

When x Sin y Cos 90 Tan180 0 Sin y 0 Then y 04

Then by substitute in 1 : Cos

2dy dy dy

0 2Sin 4Sec 0 2 4 0 -6 dx 2 dx dx

----------------------------------------------------------------------------------------- --------------


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Calculus 3rdsecondary H igher Deri vative Part II --

Example (25)2

2

2

d yIf y Cos x , Prove that : 4y 2 0

d x

Answer

2 2

dyDifferentiate with respect to x : 2Cos x -Sin x -2Sin xCos x -2Sin2x

dx

L.H.S. : - 2Cos2x 4Cos x 2 -2Cos2x 2 2Cos x 1 -2Cos2x 2Cos2x 0

----------------------------------------------------------------------------------------- --------------Example (26)

22

2

d yIf y Z 3Z 7 , Z Sin 3x , Then Prove that : at x

d x 3

Answer

2

2

2

2

22

2

2

2

dy dz dy dy dz 2Z 3 & 3Cos3x 2Z 3 3Cos3x

dz dx dx dz dx

d y dz Differentiate with respect to Z : 2Z 3 -9 Sin3x 3Cos3x 2

dx dx

d y2Z 3 -9 Sin3x 3Cos3x 2 3Cos3x

dx

d y

2Sin3x 3 -9 Sin3x 18Cos 3xdx

d yAt x : 2S

3 dx

2

22

2

in3 3 -9 Sin3 18Cos 33 3 3

d y2Sin 3 -9 Sin 18Cos 18

dx

----------------------------------------------------------------------------------------- --------------Example (27)

23 3

2 4

d y 1

If y Sin x , Z Cos x , Then Prove that : dz 3Sin xCos x Answer

2 2 2

2

2

2 2 22

2 2 2 4

dy dz 3Sin x Cos x & 3Cos x -Sin x -3Sin xCos x

dx dx

dy dy dx 3Cos x Sin-Tan x

dz dx dz -3Sin xCos x

Differentiate with respect to Z :

d y dx -Sec x Sec x 1-Sec xdz dz -3Sin xCos x 3Sin xCos x 3Sin xCos x

higher derivative

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