9.1. D - a . o h`am riˆeng 111 2. Tu . o . ng tu . . :nˆe ´ utˆo ` nta . i gi´o . iha . n lim ∆y→0 ∆ y w ∆y = lim ∆y→0 f(x, y +∆y) −f(x, y) ∆y th`ı gi´o . iha . nd ´odu . o . . cgo . il`ad a . o h`am riˆeng cu ’ a h`am f(x, y) theo biˆe ´ n y ta . idiˆe ’ m M(x, y)v`adu . o . . cchı ’ bo . ’ imˆo . t trong c´ac k´yhiˆe . u ∂w ∂y , ∂f(x, y) ∂y ,f  y (x, y),w  y . T`u . di . nh ngh˜ıa suy r˘a ` ng da . o h`am riˆeng cu ’ a h`am hai biˆe ´ n theo biˆe ´ n x l`a da . o h`am thˆong thu . `o . ng cu ’ a h`am mˆo . tbiˆe ´ n x khi cˆo ´ di . nh gi´a tri . cu ’ abiˆe ´ n y.Dod ´o c ´a c da . o h`am riˆeng du . o . . c t´ınh theo c´ac quy t˘a ´ cv`a cˆong th´u . c t´ınh da . o h`am thˆong thu . `o . ng cu ’ a h`am mˆo . tbiˆe ´ n. Nhˆa . nx´et. Ho`an to`an tu . o . ng tu . . ta c´o thˆe ’ di . nh ngh˜ıa da . o h`am riˆeng cu ’ a h`am ba (ho˘a . c nhiˆe ` uho . n ba) biˆe ´ nsˆo ´ . 9.1.2 D - a . o h`am cu ’ a h`am ho . . p Nˆe ´ u h`am w = f(x, y), x = x(t), y = y(t)th`ıbiˆe ’ uth´u . c w = f[x(t),y(t)] l`a h`am ho . . pcu ’ a t. Khi d ´o dw dt = ∂w ∂x · dx dt + ∂w ∂y · dy dt · (9.1) Nˆe ´ u w = f(x, y), trong d ´o x = x(u, v), y = y(u, v)th`ı        ∂w ∂u = ∂w ∂x ∂x ∂u + ∂w ∂y ∂y ∂u , ∂w ∂v = ∂w ∂x ∂x ∂v + ∂w ∂y ∂y ∂v · (9.2) 9.1.3 H`am kha ’ vi Gia ’ su . ’ h`am w = f(M) x´ac d i . nh trong mˆo . t lˆan cˆa . n n`ao d´ocu ’ adiˆe ’ m M(x, y). H`am f du . o . . cgo . i l`a h`am kha ’ vi ta . idiˆe ’ m M(x, y)nˆe ´ usˆo ´ gia 112 Chu . o . ng 9. Ph´ep t´ınh vi phˆan h`am nhiˆe ` ubiˆe ´ n ∆f(M)=f(x +∆,y+∆y) − f(x, y)cu ’ a h`am khi chuyˆe ’ nt`u . d iˆe ’ m M(x, y)dˆe ´ ndiˆe ’ N(x +∆,y+∆y) c´o thˆe ’ biˆe ’ udiˆe ˜ ndu . ´o . ida . ng ∆f(M)=D 1 ∆x + D 2 ∆y + o(ρ),ρ→ 0 trong d ´o ρ =  ∆x 2 +∆y 2 . Nˆe ´ u h`am f( x, y) kha ’ vi ta . id iˆe ’ m M(x, y)th`ı ∂f ∂x (M)=D 1 , ∂f ∂y (M)=D 2 v`a khi d´o ∆f(M)= ∂f ∂x (M)∆x + ∂f ∂y ∆y + o(ρ),ρ→ 0. (9.3) 9.1.4 D - a . o h`am theo hu . ´o . ng Gia ’ su . ’ : (1) w = f(M) l`a h`am x´ac di . nh trong lˆan cˆa . n n`ao d´o c u ’ adiˆe ’ m M(x, y); (2) e = (cos α, cos β) l`a vecto . do . nvi . trˆen du . `o . ng th˘a ’ ng c´o hu . ´o . ng L qua diˆe ’ m M(x, y); (3) N = N(x +∆x, y +∆y)l`adiˆe ’ m thuˆo . c L v`a ∆e l`a dˆo . d`ai cu ’ a doa . n th˘a ’ ng MN. Nˆe ´ utˆo ` nta . i gi´o . iha . nh˜u . uha . n lim ∆→0 (N→M) ∆w ∆ th`ı gi´o . iha . nd´odu . o . . cgo . il`ada . o h`am ta . idiˆe ’ m M(x, y) theo hu . ´o . ng cu ’ a vecto . e v`a du . o . . ck´yhiˆe . ul`a ∂w ∂e ,t´u . cl`a ∂w ∂e = lim ∆→0 ∆w ∆ · 9.1. D - a . o h`am riˆeng 113 Da . o h`am theo hu . ´o . ng cu ’ a vecto . e = (cos α,cos β)d u . o . . c t´ınh theo cˆong th´u . c ∂f ∂e = ∂f ∂x (M) cos α + ∂f ∂y (M) cos β. (9.4) trong d ´o cos α v`a cos β l`a c´ac cosin chı ’ phu . o . ng cu ’ a vecto . e . Vecto . v´o . i c´ac to . ad ˆo . ∂f ∂x v`a ∂F ∂y (t ´u . c l`a vecto .  ∂f ∂x , ∂f ∂y  )d u . o . . cgo . i l`a vecto . gradiˆen cu ’ a h`am f(M)ta . id iˆe ’ m M(x, y)v`adu . o . . ck´yhiˆe . ul`a gradf(M). T`u . d´o d a . o h`am theo hu . ´o . ng ∂f ∂e c´o biˆe ’ uth´u . cl`a ∂f ∂e =  gradf,e  . Ta lu . u´yr˘a ` ng: 1) Nˆe ´ u h`am w = f(x, y) kha ’ vi ta . idiˆe ’ m M(x, y) th`ı n´o liˆen tu . cta . i M v`a c´o c´ac da . o h`am riˆeng cˆa ´ p1ta . id´o ; 2) N´eu h`am w = f(x, y) c´o c´ac d a . o h`am riˆeng cˆa ´ p 1 theo mo . ibiˆe ´ n trong lˆan cˆa . nn`aod´ocu ’ adiˆe ’ m M(x, y) v`a c´ac da . o h`am riˆeng n`ay liˆen tu . cta . idiˆe ’ m M(x, y) th`ı n´o kha ’ vi ta . idiˆe ’ m M. Nˆe ´ u h`am f( x, y) kha ’ vi ta . id iˆe ’ m M(x, y) th`ı n´o c´o da . o h`am theo mo . ihu . ´o . ng ta . idiˆe ’ md´o . Ch´u´y.Nˆe ´ u h`am f(x, y)c´oda . o h`am theo mo . ihu . ´o . ng ta . idiˆe ’ m M 0 th`ı khˆong c´o g`ıda ’ mba ’ o l`a h`am f(x, y) kha ’ vi ta . idiˆe ’ m M 0 (xem v´ı du . 4). 9.1.5 D - a . o h`am riˆeng cˆa ´ p cao Gia ’ su . ’ miˆe ` n D ⊂ R 2 v`a f : D → R 114 Chu . o . ng 9. Ph´ep t´ınh vi phˆan h`am nhiˆe ` ubiˆe ´ n l`a h`am hai biˆe ´ n f(x, y)du . o . . c cho trˆen D.Tad ˘a . t D x =  (x, y) ∈ D : ∃ ∂f ∂x = ±∞  , D y =  (x, y) ∈ D : ∃ ∂f ∂y = ±∞  . D ∗ = D x ∩ D y D - i . nh ngh˜ıa. 1) C´ac da . o h`am riˆeng ∂f ∂x v`a ∂f ∂y du . o . . cgo . i l`a c´ac d a . o h`am riˆeng cˆa ´ p1. 2) Nˆe ´ u h`am ∂f ∂x : D x → R v`a ∂f ∂y : D y → R c´o c´ac da . o h`am riˆeng ∂ ∂x  ∂f ∂x  = ∂ 2 f ∂x∂x = ∂ 2 f ∂x 2 , ∂ ∂y  ∂f ∂x  = ∂ 2 f ∂x∂y , ∂ ∂x  ∂f ∂y  = ∂ 2 f ∂y∂x , ∂ ∂y  ∂f ∂y  = ∂ 2 f ∂y∂y = ∂ 2 f ∂y 2 th`ı ch´ung du . o . . cgo . i l`a c´ac da . o h`am riˆeng cˆa ´ p2theo x v`a theo y. C´ac da . o h`am riˆeng cˆa ´ p3du . o . . cdi . nh ngh˜ıa nhu . l`a c´ac da . o h`am riˆeng cu ’ ada . o h`am riˆeng cˆa ´ p 2, v.v Ta lu . u´yr˘a ` ng nˆe ´ u h`am f(x, y) c´o c´ac da . o h`am hˆo ˜ nho . . p ∂ 2 f ∂x∂y v`a ∂ 2 f ∂y∂x liˆen tu . cta . idiˆe ’ m(x,y) th`ı ta . idiˆe ’ md´o c´ac da . o h`am hˆo ˜ nho . . p n`ay b˘a ` ng nhau: ∂ 2 f ∂x∂y = ∂ 2 f ∂y∂x · C ´ AC V ´ IDU . 9.1. D - a . o h`am riˆeng 115 V´ı du . 1. T´ınh da . o h`am riˆeng cˆa ´ p1cu ’ a c´ac h`am 1) 4w = x 2 − 2xy 2 + y 3 .2)w = x y . Gia ’ i. 1) D a . o h`am riˆeng ∂w ∂x du . o . . c t´ınh nhu . l`a d a . o h`am cu ’ a h`am w theo biˆe ´ n x v´o . i gia ’ thiˆe ´ t y = const. Do d´o ∂w ∂x =(x 2 − 2xy 2 + y 3 )  x =2x − 2y 2 +0=2(x −y 2 ). Tu . o . ng tu . . , ta c´o ∂w ∂y =(x 2 −2xy 2 + y 3 )  y =0−4xy +3y 2 = y(3y − 4x). 2) Nhu . trong 1), xem y = const ta c´o ∂w ∂x =  x y   x = yx y−1 . Tu . o . ng tu . . , khi xem x l`a h˘a ` ng sˆo ´ ta thu d u . o . . c ∂w ∂y = x y lnx. (v`ı w = x y l`a h`am m˜udˆo ´ iv´o . ibiˆe ´ n y khi x = const.  V´ı d u . 2. Cho w = f(x, y)v`ax = ρ cos ϕ, y = ρ sin ϕ. H˜ay t´ınh ∂w ∂ρ v`a ∂w ∂ϕ . Gia ’ i. D ˆe ’ ´ap du . ng cˆong th´u . c (9.2), ta lu . u´yr˘a ` ng w = f( x, y)=f(ρ cos ϕ, ρ sin ϕ)=F (ρ, ϕ). Do d´o theo (9.2) v`a biˆe ’ uth´u . cdˆo ´ iv´o . i x v`a y ta c´o ∂w ∂ρ = ∂w ∂x ∂x ∂ρ + ∂w ∂y ∂y ∂ρ = ∂w ∂x cos ϕ + ∂w ∂y sin ϕ ∂w ∂ϕ = ∂w ∂x ∂x ∂ϕ + ∂w ∂y ∂y ∂ϕ = ∂w ∂x (−ρ sin ϕ)+ ∂w ∂y (ρ cos ϕ) = ρ  − ∂w ∂x sin ϕ + ∂w ∂y cos ϕ  .  116 Chu . o . ng 9. Ph´ep t´ınh vi phˆan h`am nhiˆe ` ubiˆe ´ n V´ı du . 3. T´ınh da . o h`am cu ’ a h`am w = x 2 + y 2 x ta . idiˆe ’ m M 0 (1, 2) theo hu . ´o . ng cu ’ a vecto . −→ M 0 M 1 , trong d´o M 1 l`a diˆe ’ mv´o . ito . ad ˆo . (3, 0). Gia ’ i. D ˆa ` u tiˆen ta t`ım vecto . d o . nvi . e c´o hu . ´o . ng l`a hu . ´o . ng d ˜a cho. Ta c´o −→ M 0 M 1 =(2,−2)=2e 1 − 2e 2 , ⇒| −→ M 0 M 1 | =2 √ 2 ⇒ e = M 0 M 1 |M 0 M 1 | = 2e 1 − 2e 2 2 √ 2 = 1 √ 2 e 1 − 1 √ 2 e 2 . trong d ´o e 1 , e 2 l`a vecto . do . nvi . cu ’ a c´ac tru . cto . adˆo . .T`u . d´o suy r˘a ` ng cos α = 1 √ 2 , cos β = − 1 √ 2 · Tiˆe ´ p theo ta t´ınh c´ac d a . o h`am riˆeng ta . idiˆe ’ m M 0 (1, 2). Ta c´o f  x =2x + y 2 ⇒ f  x (M 0 )=f  x (1, 2)=6, f  y =2xy ⇒ f  y (M 0 )=f  y (1, 2)=4. Do d ´o theo cˆong th´u . c (9.4) ta thu d u . o . . c ∂f ∂e =6· 1 √ 2 − 4 · 1 √ 2 = √ 2.  V´ı d u . 4. H`am f( x, y)=x + y +  |xy| c´o da . o h`am theo mo . ihu . ´o . ng ta . idiˆe ’ m O(0, 0) nhu . ng khˆong kha ’ vi ta . id´o. Gia ’ i. 1. Su . . tˆo ` nta . id a . o h`am theo mo . ihu . ´o . ng. Ta x´et hu . ´o . ng cu ’ a vecto . e d irat`u . Ov`alˆa . pv´o . i tru . c Ox g´oc α.Ta c´o ∆ e f(0, 0) = ∆x +∆y +  |∆x∆y| =  cos α + sin α +  |cos α sin α|  ρ, 9.1. D - a . o h`am riˆeng 117 trong d´o ρ =  ∆x 2 +∆y 2 ,∆x = ρ cos α,∆y = ρ sin α. T`u . d ´o suy ra ∂f ∂e (0, 0) = lim ρ→0 ∆ e f(0, 0) ρ = cos α + sin α +  |sin α cos α| t´u . cl`ad a . o h`am theo hu . ´o . ng tˆo ` nta . i theo mo . ihu . ´o . ng. 2. Tuy nhiˆen h`am d ˜a cho khˆong kha ’ vi ta . i O. Thˆa . tvˆa . y, ta c´o ∆f(0, 0) = f(∆x, ∆y) −f(0 , 0)=∆x +∆y +  |∆x||∆y|−0. V`ı f  x =1v`af  y = 1 (ta . i sao ? ) nˆen nˆe ´ u f kha ’ vi ta . i O(0, 0) th`ı ∆f(0, 0) = ∆x +∆y +  |∆x∆y| =1·∆x +1· ∆y + ε(ρ)ρ ε(ρ) → 0(ρ → 0),ρ=  ∆x 2 +∆y 2 hay l`a lu . u´y∆x = ρ cos α,∆y = ρ sin α ta c´o ε(ρ)=  |cos α sinα|. Vˆe ´ pha ’ id˘a ’ ng th´u . c n`ay khˆong pha ’ i l`a vˆo c`ung b´e khi ρ → 0 (v`ı n´o ho`an to`an khˆong phu . thuˆo . c v`ao ρ). Do d ´o theo di . nh ngh˜ıa h`am f(x, y) d˜a cho khˆong kha ’ vi ta . idiˆe ’ mO. V´ı du . 5. T´ınh c´ac d a . o h`am riˆeng cˆa ´ p2cu ’ a c´ac h`am: 1) w = x y ,2)w = arctg x y · Gia ’ i. 1) D ˆa ` u tiˆen t´ınh c´ac da . o h`am riˆeng cˆa ´ p1.Tac´o ∂w ∂x = yx y−1 , ∂w ∂y = x y lnx. Tiˆe ´ p theo ta c´o ∂ 2 w ∂x 2 = y(y −1)x y−2 , ∂ 2 w ∂y∂x = x y−1 + yx y−1 lnx = x y−1 (1 + ylnx), ∂ 2 w ∂x∂y = yx y−1 lnx + x y · 1 x = x y−1 (1 + ylnx), ∂ 2 f ∂y 2 = x y (lnx) 2 . 118 Chu . o . ng 9. Ph´ep t´ınh vi phˆan h`am nhiˆe ` ubiˆe ´ n 2) Ta c´o ∂w ∂x = y x 2 + y 2 , ∂w ∂y = − x x 2 + y 2 · T`u . d ´o ∂ 2 w ∂x 2 = ∂ ∂x  y x 2 + y 2  = − 2xy (x 2 + y 2 ) 2 , ∂ 2 w ∂y 2 = ∂ ∂y  −x x 2 + y 2  = 2xy x 2 + y 2 , ∂ 2 w ∂x∂y = ∂ ∂y  y x 2 + y 2  = x 2 − y 2 (x 2 + y 2 ) 2 , ∂ 2 w ∂y∂x = ∂ ∂x  − x x 2 + y 2  = x 2 −y 2 (x 2 + y 2 ) 2 · Nhˆa . nx´et. Trong ca ’ 1) lˆa ˜ n2)tadˆe ` uc´o ∂ 2 w ∂x∂y = ∂ 2 w ∂y∂x .  V´ı d u . 6. T´ınh c´ac da . o h`am riˆeng cˆa ´ p1cu ’ a h`am w = f(x+y 2 ,y+x 2 ) ta . id iˆe ’ m M 0 (−1, 1), trong d´o x v`a y l`a biˆe ´ ndˆo . clˆa . p. Gia ’ i. D ˘a . t t = x + y 2 , v = y + x 2 . Khi d´o w = f( x + y 2 ,y+ x 2 )=f(t, v). Nhu . vˆa . y w = f( t, v) l`a h`am ho . . pcu ’ a hai biˆe ´ ndˆo . clˆa . p x v`a y. N´o phu . thuˆo . c c´ac biˆe ´ ndˆo . clˆa . p thˆong qua hai biˆe ´ n trung gian t, v. Theo cˆong th ´u . c (9.2) ta c´o: ∂w ∂x = ∂f ∂t · ∂t ∂x + ∂f ∂v · ∂v ∂x = f  t (x + y 2 ,y+ x 2 ) ·1+f  v (x + y 2 ,y+ x 2 ) ·2x = f  t +2xf  v . 9.1. D - a . o h`am riˆeng 119 ∂w ∂x (−1, 1) = ∂f ∂x (0, 2) = f  t (0, 2) −2f  v (0, 2) ∂w ∂y = ∂f ∂t · ∂t ∂y + ∂f ∂v · ∂v ∂y = f  t (·)2y + f  v (·)1 =2yf  t + f  v ∂w ∂y (−1, 1) = ∂f ∂y (0, 2)=2f  t (0, 2) + f  v (0, 2).  B ` AI T ˆ A . P T´ınh d a . o h`am riˆeng cu ’ a c´ac h`am sau dˆay 1. f(x, y)=x 2 + y 3 +3x 2 y 3 . (DS. f  x =2x +6xy 3 , f  y =3y 2 +9x 2 y 2 ) 2. f(x, y, z)=xyz + x yz . (DS. f  x = yz + 1 yz , f  y = xz − x y 2 z , f  z = xy − x yz 2 ) 3. f(x, y, z) = sin(xy + yz). (D S. f  x = y cos(xy + yz), f  y =(x + z) cos(xy + yz), f  z = y cos(xy + yz)) 4. f(x, y) = tg(x + y)e x/y . (D S. f  x = e x/y cos 2 (x + y) + tg(x + y)e x/y 1 y , f  y = e x/y cos 2 (x + y) + tg(x + y)e x/y  − x y 2  .) 5. f = arc sin x  x 2 + y 2 .(DS. f  x = |y| x 2 + y 2 , f  y = −xsigny x 2 + y 2 ) 6. f(x, y)=xyln(xy). (DS. f  x = yln(xy)+y, f  y = xln(xy)+x) 120 Chu . o . ng 9. Ph´ep t´ınh vi phˆan h`am nhiˆe ` ubiˆe ´ n 7. f( x, y, z)=  y x  z . (DS. f  x = z  y x  z−1  − y x 2  = − z x  y x  z , f  y = z y  y x  z ,f  z =  y x  z ln y x ) 8. f( x, y, z)=z x/y . (D S. f  x = x x/y lnz ·  1 y  , f  y = z x/y lnz ·  −x y 2  , f  z =  x y  z x/y−1 ) 9. f( x, y, z)=x y z . (DS. f  x = y z x y z −1 , f  y = x y z zy z−1 lnx, f  z = x y z ln(x) z lny) 10. f( x, y, z)=x y y z z x . (DS. f  x = x y−1 y z+1 z x + x y y z z x lnz, f  y = x y lnxy z z x + x y y z−1 z x+1 , f  z = x y y z lny · z x + x y+1 y z z x−1 ) 11. f( x, y) = ln sin x + a √ y . (D S. f  x = 1 √ y cotg x + a √ y , f  y = − x + a y cotg x + a √ y ) 12. f( x, y)= x y − e x arctgy. (DS. f  x = 1 y − e x arctgy, f  y = − x y 2 − e x 1+y 2 ) 13. f( x, y)=ln  x +  x 2 + y 2  . (D S. f  x = 1  x 2 + y 2 , f  y = 1 x +  x 2 + y 2 · y  x 2 + y 2 ). T`ım da . o h`am riˆeng cu ’ a h`am ho . . p sau d ˆay (gia ’ thiˆe ´ t h`am f(x, y) kha ’ vi) 14. f( x, y)=f(x + y, x 2 + y 2 ). (DS. f  x = f  t + f  v 2x, f  y = f  t + f  v 2y, t = x + y, v = x 2 + y 2 ) 15. f( x, y)=f  x y , y x  . [...]... hu.´.ng gradien cua h`m d´ (DS ) 5 ´.ng cua vecto gradien cua h`m ’ ’ a 53 T` gi´ tri v` hu o ım a a w = tgx − x + 3 sin y − sin3 y + z + cotgz π π π ’ tai diˆm M0 , , e 4 3 2 3 8 3 (DS (gradw)M = i + j, cos α = √ , cos β = √ ) 8 73 73 z ’ ’ a tai diˆm M0 (1, 1, 1) 54 T` dao h`m cua h`m w = arc sin ım a e x2 + y 2 −→ 1 o o theo hu.´.ng vecto M0 M, trong d´ M = (3, 2, 3) (DS ) 6 9.2 ´ ` ’ Vi phˆn cua... fxy = −z sin t, fxz = −y sin t, fyy = −z 2 sin t, fyz = −yz sin t, fzz = −y 2 sin t, t = x + yz) ∂ 2f ´ nˆu f = x2 + y 2 ex+y e ∂x∂y ex+y − xy + (x + y)(x2 + y 2) + (x2 + y 2)2 ) (DS 2 2 )3/2 (x + y 28 T´ ınh ∂ 2f ∂ 2f ∂ 2f ´ , , nˆu f = xyz e ∂x∂y ∂y∂z ∂x∂z (DS fxy = xyz−1 z(1 + yzlnx), fxz = xyz−1 y(1 + yzlnx), fyz = lnx · xyz (1 + yzlnx)) 29 T´ ınh x+y ∂ 2f ∂ 2f ´ nˆu f = arctg e (DS = 0) 30 T´... x2 ftt − 2 2 ftv + 4 fvv + 3 fv , y y y x t = xy, v = ) y uxy = xyftt − uyy 37 u = f (sin x + cos y) (DS uxx = cos2 x · f − sin x · f , uxy = − sin y cos x · f , uyy = sin2 y · f − cos y · f ) ` a a 38 Ch´.ng minh r˘ng h`m u f= (x−x0 )2 1 √ e− 4a2 t 2a πt ` ´ ’ ınh e e (trong d´ a, x0 l` c´c sˆ) thoa m˜n phu.o.ng tr` truyˆn nhiˆt o a a o a ∂ 2f ∂f = a2 2 · ∂t ∂x ´ ` Chu.o.ng 9 Ph´p t´ vi phˆn h`m... T` dao h`m cua h`m f (x, y) = ln x2 + y 2 tai diˆm M(1, 1) ım a a e √ 2 ` ´ o a a ’ o a u a ) theo hu.´.ng phˆn gi´c cua g´c phˆn tu th´ nhˆt (DS 2 ´ ` 9.2 Vi phˆn cua h`m nhiˆu biˆn a ’ a e e ’ ’ 48 T` dao h`m cua h`m f (x, y, z) = z 2 − 3xy + 5 tai diˆm ım a a e ´.ng lˆp v´.i c´c truc toa dˆ nh˜.ng g´c b˘ng nhau ` a o a o a M(1, 2,√ theo hu o −1) o u 3 (DS − ) 3 ´ ’ a 49 T` dao h`m cua h`m... y) = f (x − y, xy) (DS fx = ft + yfv , fy = −ft + xfv , t = x − y, v = xy) 17 f (x, y) = f (x − y 2, y − x2 , xy) (DS fx = ft − 2xfv + yfw , fy = −2yft + fv + xfw , t = x − y 2, v = y − x2, w = xy) √ 18 f (x, y, z) = f ( x2 + y 2 , y 2 + z 2, z 2 + x2 ) (DS fx = fz = v= xft xfw yft yf , fy = +√ v , z 2 + x2 x2 + z 2 x2 + y 2 x2 + y 2 zfv zf + √ w , t = x2 + y 2 , 2 + y2 z 2 + x2 x √ y 2 + z 2 , w = . − sin 3 y + z + cotgz ta . idiˆe ’ m M 0  π 4 , π 3 , π 2  . (D S. (gradw) M =  i + 3 8  j, cos α = 8 √ 73 , cos β = 3 √ 73 ) 54. T`ım da . o h`am cu ’ a h`am w = arc sin z  x 2 + y 2 ta . idiˆe ’ m. x y−1 (1 + ylnx), ∂ 2 w ∂x∂y = yx y−1 lnx + x y · 1 x = x y−1 (1 + ylnx), ∂ 2 f ∂y 2 = x y (lnx) 2 . 1 18 Chu . o . ng 9. Ph´ep t´ınh vi phˆan h`am nhiˆe ` ubiˆe ´ n 2) Ta c´o ∂w ∂x = y x 2 + y 2 , ∂w ∂y =. z  y x  z−1  − y x 2  = − z x  y x  z , f  y = z y  y x  z ,f  z =  y x  z ln y x ) 8. f( x, y, z)=z x/y . (D S. f  x = x x/y lnz ·  1 y  , f  y = z x/y lnz ·  −x y 2  , f  z =  x y  z x/y−1 ) 9.