ISM_T11_C03_C - 160 Chapter 3 Differentiation 3.6 IMPLICIT...

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160 Chapter 3 Differentiation 3.6 IMPLICIT DIFFERENTIATION 1. y x x 2. y x x œÊ œ œ Ê œ ± *Î% &Î% ±$Î& ±)Î& dy dy dx 4 dx 5 9 3 3. y 2x (2x) (2x) 2 4. y 5x (5x) (5x) 5 œœ Ê œ œ Ê œ œ $ % "Î$ ±#Î$ "Î% ±$Î% " " È È dy dy dx 3 dx 4 2 5 3x 4x †† "Î$ "Î% #Î$ $Î% 5. y 7 x 6 7(x 6) (x 6) œ² œ ² Ê œ ² œ È "Î# ±"Î# # ² dy dx 77 2x6 È 6. y 2 x 1 2(x 1) 1(x 1) œ ± ± œ ±± Ê œ œ ± È "Î# ±"Î# " ± dy dx x1 È 7. y (2x 5) (2x 5) 2 (2x 5) Êœ ± ² œ ± ² ±"Î# ±$Î# ±$Î# " # dy dx 8. y ( 6x) (1 6x) ( 6) 4(1 6x) œ"± Ê œ ± ± œ± ± #Î$ ±"Î$ ±"Î$ dy dx 3 2 9. y x x 1 y x x 1 x x 1 x 1 x x Ê œ †² # ² ²† " œ ² ² ² " œ ab a b a b #w # # # # # "Î# ±"Î# "Î# ±"Î# " ² # ² 2x 1 # # È 10. y x x 1 y x x 1 x x 1 x 1 x x œ ² Ê œ † ± ² # ² ² †"œ ² ± ² ²" œ a b a b ˆ‰ # # # # # ±"Î# ±$Î# ±"Î# ±$Î# " " # ² # $Î# 11. s t t t 12. r Ê œ œ œ Ê œ ± ( % # ±$ #Î( ±&Î( ±$Î% ±(Î% ÈÈ ds 2 dr 3 dt 7 d4 )) ) ) 13. y sin (2t 5) cos (2t 5) (2t 5) 2 (2t 5) cos (2t 5) Ê œ ² ± ² œ ± ² ² ˆ ±#Î$ ±#Î$ ±&Î$ ±&Î$ ±#Î$ dy dt 3 3 24 14. z cos ( 6t) sin ( 6t) (1 6t) ( ) 4(1 6t) sin (1 6t) œ" ± Ê œ ± " ± ± ± ' œ ± ± #Î$ #Î$ ±"Î$ ±"Î$ #Î$ dz 2 dt 3 15. f(x) 1 x 1 x f (x) 1 x x œ±œ ± Ê œ ± ± œ œ É È ˆ "Î# w "Î# ±"Î# "Î# ±"Î# " " ±" ±" ## ± ± 41 xx 4x1 x Š‹ ÉÉ È 16. g(x) 2 2x 1 g (x) 2x 1 ( 1)x 2x 1 x Ê œ ±² ± ±"Î# w ±"Î# ±$Î# ±"Î# ±$Î# ±"Î$ ±%Î$ ±%Î$ 22 33 17. h( ) 1 cos (2 ) (1 cos 2 ) h ( ) (1 cos 2 ) ( sin 2 ) 2 (sin 2 )(1 cos 2 ) ) ) ) ) œ ² Ê ± œ ± ² $ "Î$ w ±#Î$ ±#Î$ " È 2 18. k( ) (sin ( 5)) k ( ) (sin ( 5)) cos ( 5) cos ( 5)(sin ( 5)) ) ) ) ) ) œ²Ê œ ² ² œ ² ² &Î% w "Î% "Î% 55 44 19. x y xy 6: ²œ Step 1: x y 2x x 2y y 1 0 Š dy dy dx dx ²² Step 2: 2xy 2xy y dy dy dx dx ± ± Step 3: x 2xy 2xy y dy dx ± ± Step 4: dy 2xy y dx x 2xy œ ² # # 20. x y 18xy 3x 3y 18y 18x 3y 18x 18y 3x $$ # # # # ± ± ² œ ʲ Ê ± œ±Ê œ dy dy dy dy 6y x dx dx dx dx y 6x # # 21. 2xy y x y: ²œ² # Step 1: 2x 2y 2y 1 dy dy dy dx dx dx œ ²
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Section 3.6 Implicit Differentiation 161 Step 2: 2x 2y 1 2y dy dy dy dx dx dx ±² œ ² S t e p 3 : ( 2 x2 y1 ) 2 y dy dx ±²œ " ² Step 4: dy 1 2y dx 2x 2y 1 œ ± ²± 22. x xy y 1 3x y x 3y 0 3y x y 3x $$ # # # # ± ± ²±œÊ ²² ± œÊ ² œ² Ê œ dy dy dy dy y 3x dx dx dx dx 3y x ab # # 23. x (x y) x y : ## # # ²œ ² Step 1: x 2(x y) 1 (x y) (2x) 2x 2y ’“ Š‹ ²²± ² œ ² dy dy dx dx Step 2: 2x (x y) 2y 2x 2x (x y) 2x(x y) œ ² ² ²² # dy dy dx dx Step 3: 2x (x y) 2y 2x 1 x(x y) (x y) dy dx cd c d ² ² ±œ ² ² Step 4: dy dx 2x (x y) 2y y x (x y) x y x y 2x 1 x(x y) (x y) x 1 x(x y) (x y) x 1 x xy x 2xy y œœ œ ± ±±± ±± ² ± ± ± ² ±²±² ± # $ # œ x 3 x yx y xy x y ±² ± $# # #$ 24. (3xy 7) 6y 2(3xy 7) 3x 3y 6 2(3xy 7)(3x) 6 6y(3xy 7) ± œ ʱ ± œ ² œ # dy dy dy dy dx dx dx dx [6x(3xy 7) 6] 6y(3xy 7) ² œ ² ± Ê œ dy dy y(3xy 7) 3xy 7y dx dx x(3xy 7) 1 1 3x y 7x ± ± # # 25. y 2y # ±" " ² ² ²±± œÊ œ œ Ê œ x 1 dx (x 1) (x 1) dx y(x 1) dy (x 1) (x 1) dy # 26. x x x y x y 3x 2xy x y 1 y x 1 y 1 3x 2xy y # # # w w # w # w ± ² ² œ Ê ± œ² Ê ± ± œ² Ê ± œ² ² Ê œ xy 1 3x 2xy x1 # # 27. x tan y 1 sec y cos y œ Ê œ œ " dy dy dx dx sec y # 28. xy cot xy x y csc (xy) x x x csc (xy) y csc (xy) y ± œ ² ± Ê ± œ ² ² dy dy
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This note was uploaded on 09/20/2010 for the course MATHEMATIC 09991051 taught by Professor Dr.maenshadeed during the Fall '10 term at Norwegian Univ. of Science & Technology.

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ISM_T11_C03_C - 160 Chapter 3 Differentiation 3.6 IMPLICIT...

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