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8.1. D - a . o h`am 63 f(x) f (x) f (n) (x) cos x −sin x cos x + nπ 2 tgx 1 cos 2 x cotgx − 1 sin 2 x arc sin x 1 √ 1 − x 2 , |x| < 1 arccosx − 1 √ 1 − x 2 , |x| < 1 arctgx 1 1+x 2 arccotgx − 1 1+x 2 Viˆe . c t´ınh da . o h`am du . o . . cdu . . a trˆen c´ac quy t˘a ´ c sau dˆay. 1 + d dx [u + v]= d dx u + d dx v. 2 + d dx (αu)=α du dx , α ∈ R. 3 + d dx (uv)=v du dx + u dv dx . 4 + d dx u v = 1 v 2 v du dx − u dv dx , v =0. 5 + d dx f[u(x)] = df du · du dx (da . o h`am cu ’ a h`am ho . . p). 6 + Nˆe ´ u h`am y = y(x) c´o h`am ngu . o . . c x = x(y)v`a dy dx ≡ y x =0th`ı dx dy ≡ x y = 1 y x · 64 Chu . o . ng 8. Ph´ep t´ınh vi phˆan h`am mˆo . tbiˆe ´ n 7 + Nˆe ´ u h`am y = y(x)du . o . . c cho du . ´o . ida . ng ˆa ’ nbo . ’ ihˆe . th ´u . c kha ’ vi F (x, y)=0v`aF y =0th`ı dy dx = − F x F y trong d´o F x v`a F y l`a da . o h`am theo biˆe ´ ntu . o . ng ´u . ng cu ’ a h`am F (x, y) khi xem biˆe ´ n kia khˆong dˆo ’ i. 8 + Nˆe ´ u h`am y = y(x)du . o . . cchodu . ´o . ida . ng tham sˆo ´ x = x(t), y = y(t)(x (t) = 0) th`ı dy dx = y (t) x (t) · 9 + d n dx n (αu + βv)=α d n u dx n + β d n v dx n ; d n dx n uv = n k=0 C k n d n−k dx n−k u d k dx k v (quy t˘a ´ c Leibniz). Nhˆa . nx´et. 1) Khi t´ınh da . o h`am cu ’ amˆo . tbiˆe ’ uth´u . cd˜a cho ta c´o thˆe ’ biˆe ´ nd ˆo ’ iso . bˆo . biˆe ’ uth´u . cd´o sao cho qu´a tr`ınh t´ınh da . oh`amdo . n gia ’ n ho . n. Ch˘a ’ ng ha . nnˆe ´ ubiˆe ’ uth´u . cd´o l`a logarit th`ı c´o thˆe ’ su . ’ du . ng c´ac t´ınh chˆa ´ tcu ’ a logarit d ˆe ’ biˆe ´ ndˆo ’ i rˆo ` it´ınhda . o h`am. Trong nhiˆe ` u tru . `o . ng ho . . p khi t´ınh da . o h`am ta nˆen lˆa ´ y logarit h`am d˜a cho rˆo ` i´ap du . ng cˆong th´u . cda . o h`am loga d dx lny(x)= y (x) y(x) · 2) Nˆe ´ u h`am kha ’ vi trˆen mˆo . t khoa ’ ng du . o . . cchobo . ’ iphu . o . ng tr`ınh F (x, y)=0th`ıd a . o h`am y (x) c´o thˆe ’ t`ım t`u . phu . o . ng tr`ınh d dx F (x, y)=0. C ´ AC V ´ IDU . 8.1. D - a . o h`am 65 V´ı du . 1. T´ınh da . o h`am y nˆe ´ u: 1) y =ln 3 e x 1 + cos x ; x = π(2n + 1), n ∈ N 2) y = 1+x 2 3 √ x 4 sin 7 x , x = πn, n ∈ N. Gia ’ i. 1) Tru . ´o . chˆe ´ ttad o . n gia ’ nbiˆe ’ uth´u . ccu ’ a h`am y b˘a ` ng c´ach du . . a v`ao c´ac t´ınh chˆa ´ tcu ’ a logarit. Ta c´o y = 1 3 lne x − 1 3 ln(1 + cos x)= x 3 − 1 3 ln(1 + cos x). Do d´o y = 1 3 − 1 3 (cos x) 1 + cos x = 1 3 + 1 3 sin x 1+cosx = 1+tg x 2 3 · 2) O . ’ d ˆay tiˆe . nlo . . iho . nca ’ l`a x´et h`am z =ln|y|.Tac´o dz dx = dz dy · dy dx = 1 y dy dx ⇒ dy dx = y dz dx · (*) Viˆe ´ t h`am z du . ´o . ida . ng x =ln|y| = ln(1 + x 2 ) − 4 3 ln|x|−7ln|sin x| ⇒ dz dx = 2x 1+x 2 − 4 3x − 7 cos x sin x · Thˆe ´ biˆe ’ uth´u . cv`u . athudu . o . . cv`ao(∗) ta c´o dy dx = 1+x 2 3 √ x 4 sin 7 x 2x 1+x 2 − 4 3x − 7 cos x sin x . V´ı du . 2. T´ınh d a . o h`am y nˆe ´ u: 1) y = (2+cos x) x , x ∈ R;2)y = x 2 x , x>0. Gia ’ i. 1) Theo di . nh ngh˜ıa ta c´o y = e xln(2+cosx) . 66 Chu . o . ng 8. Ph´ep t´ınh vi phˆan h`am mˆo . tbiˆe ´ n T`u . d ´o y = e xln(2+cosx) xln(2 + cos x) = e xln(2+cosx) ln(2 + cos x) − x sin x 2 + cos x ,x∈ R. 2) V`ı y = e 2 x lnx nˆen v´o . i x>0 ta c´o y = e 2 x lnx [2 x lnx] = e 2 x lnx 1 x 2 x +2 x ln2 · lnx =2 x x 2 x 1 x +ln2· lnx . V´ı du . 3. T´ınh da . o h`am cˆa ´ p2cu ’ a h`am ngu . o . . cv´o . i h`am y = x + x 5 , x ∈ R. Gia ’ i. H`am d˜a cho liˆen tu . cv`ado . ndiˆe . u kh˘a ´ pno . i, da . o h`am y = 1+5x 4 khˆong triˆe . t tiˆeu ta . ibˆa ´ tc´u . d iˆe ’ m n`ao. Do d´o x y = 1 y x = 1 1+5x 4 · Lˆa ´ yd a . o h`am d˘a ’ ng th´u . c n`ay theo y ta thu d u . o . . c x yy = 1 1+5x 4 x · x y = −20x 3 (1 + 5x 4 ) 3 · V´ı d u . 4. Gia ’ su . ’ h`am y = f(x)du . o . . cchodu . ´o . ida . ng tham sˆo ´ bo . ’ i c´ac cˆong th´u . c x = x(t), y = y(t), t ∈ (a; b) v`a gia ’ su . ’ x(t), y(t) kha ’ vi cˆa ´ p 2v`ax (t) =0t ∈ (a, b). T`ım y xx . Gia ’ i. Ta c´o dy dx = dy dt dx dt = y t x t ⇒ y x = y t x t · Lˆa ´ yd a . o h`am hai vˆe ´ cu ’ ad˘a ’ ng th ´u . c n`ay ta c´o y xx = y t x t t · t x = y t x t t · 1 x t = x t y tt − y t x tt x t 3 · 8.1. D - a . o h`am 67 V´ı d u . 5. Gia ’ su . ’ y = y(x), |x| >al`a h`am gi´a tri . du . o . ng cho du . ´o . i da . ng ˆa ’ nbo . ’ iphu . o . ng tr`ınh x 2 a 2 − y 2 b 2 =1. T´ınh y xx . Gia ’ i. Dˆe ’ t`ım y ta ´ap du . ng cˆong th´u . c d dx F (x, y)=0. Trong tru . `o . ng ho . . p n`ay ta c´o d dx x 2 a 2 − y 2 b 2 − 1 =0. Lˆa ´ yda . o h`am ta c´o 2x a 2 − 2y b 2 y x =0, (8.1) ⇒y x = b 2 x a 2 y , |x| > 0,y >0. (8.2) Lˆa ´ yd a . o h`am (8.1) theo x ta thu du . o . . c 1 a 2 − 1 b 2 y x 2 − y b 2 y xx =0 v`a t`u . (8.2) ta thu du . o . . c y x : y xx = 1 y b 2 a 2 − y x 2 = 1 y b 2 a 2 − b 4 a 4 x 2 y 2 = − b 4 a 2 y 3 x 2 a 2 − y 2 b 2 = − b 4 a 2 y 3 ,y>0. V´ı du . 6. T´ınh y (n) nˆe ´ u: 1) y = 1 x 2 − 4 ;2)y = x 2 cos 2x. Gia ’ i. 1) Biˆe ’ udiˆe ˜ nh`amd˜achodu . ´o . ida . ng tˆo ’ ng c´ac phˆan th´u . cco . ba ’ n 1 x 2 − 4 = 1 4 1 x − 2 − 1 x +2 68 Chu . o . ng 8. Ph´ep t´ınh vi phˆan h`am mˆo . tbiˆe ´ n v`a khi d´o 1 x 2 −4 (n) = 1 4 1 x − 2 (n) − 1 x +2 (n) . Do 1 x ± 2 (n) =(−1)(−2) ···(−1 − n + 1)(x ± 2) −1−n =(−1) n n! 1 (x ± 2) n+1 nˆen 1 x 2 − 4 (n) = (−1) n n! 4 1 (x − 2) n+1 − 1 (x +2) n+1 . 2) Ta ´ap du . ng cˆong th´u . c Leibniz d ˆo ´ iv´o . id a . o h`am cu ’ at´ıch (x 2 cos 2x)=C 0 n x 2 (cos 2x) (n) + C 1 n (x 2 ) (cos 2x) n−1 + C 2 n (x 2 ) (cos 2x) n−2 . C´ac sˆo ´ ha . ng c`on la . idˆe ` u=0v`ı x 2 (k) =0 ∀k>2. ´ Ap du . ng cˆong th´u . c (cos 2x) (n) =2 n cos 2x + nπ 2 ta thu du . o . . c (x 2 cos 2x) (n) =2 n x 2 − n(n − 1) 4 cos 2x + nπ 2 +2 n nx sin 2x + nπ 2 . V´ı d u . 7. V´o . i gi´a tri . n`ao cu ’ a a v`a b th`ı h`am f(x)= e x ,x 0, x 2 + ax + b, x > 0 8.1. D - a . o h`am 69 c´o da . o h`am trˆen to`an tru . csˆo ´ . Gia ’ i. R˜o r`ang l`a h`am f(x)c´od a . o h`am ∀x>0v`a∀x<0. Ta chı ’ cˆa ` nx´etd iˆe ’ m x 0 =0. V`ı h`am f(x) pha ’ i liˆen tu . cta . id iˆe ’ m x 0 =0nˆen lim x→0+0 f(x) = lim x→0−0 f(x) = lim x→0 f(x) t´u . cl`a lim x→0+0 (x 2 + ax + b)=b = e 0 =1⇒ b =1. Tiˆe ´ pd ´o, f + (0) = (x + ax + b) x 0 =0 = a v`a f − (0) = e x x 0 =0 =1. Do d´o f (0) tˆo ` nta . inˆe ´ u a =1v`ab = 1. Nhu . vˆa . yv´o . i a =1,b =1 h`am d ˜a cho c´o da . o h`am ∀x ∈ R. B ` AI T ˆ A . P T´ınh d a . o h`am y cu ’ a h`am y = f(x)nˆe ´ u: 1. y = 4 √ x 3 + 5 x 2 − 3 x 3 + 2. (DS. 3 4 4 √ x − 10 x 3 + 9 x 4 ) 2. y = log 2 x + 3log 3 x.(DS. ln24 xln2 · ln3 ) 3. y =5 x +6 x + 1 7 x .(DS. 5 x ln5 + 6 x ln6 − 7 −x ln7) 4. y = ln(x +1+ √ x 2 +2x + 3). (DS. 1 √ x 2 +2x +3 ) 5. y = tg5x.(DS. 10 sin 10x ) 6. y = ln(ln √ x). (DS. 1 2xln √ x ) 7. y =ln 1+2x 1 − 2x .(DS. 2 1 − 4x 2 ) 70 Chu . o . ng 8. Ph´ep t´ınh vi phˆan h`am mˆo . tbiˆe ´ n 8. y = xarctg √ 2x − 1 − √ 2x − 1 2 .(DS. arctg √ 2x − 1) 9. y = sin 2 x 3 .(DS. 3x 2 sin 2x 3 ) 10. y = sin 4 x + cos 4 x.(DS. −sin 4x) 11. y = √ xe √ x .(DS. e √ x (1 + √ x) 2 √ x ) 12. y = e 1 cos x .(D S. e 1 cos x sin x cos 2 x ) 13. y = e 1 lnx .(DS. −e 1 lnx xln 2 x ) 14. y =ln e 2x + √ e 4x +1. (DS. 2e 2x √ e 4x +1 ) 15. y =ln e 4x e 4x +1 .(DS. 2 e 4x +1 ) 16. y = log 5 cos 7x.(DS. − 7tg7x ln5 ) 17. y = log 7 cos √ 1+x.(DS. − tg √ 1+x 2 √ 1+xln7 ) 18. y = arccos e − x 2 2 .(D S. xe − x 2 2 √ 1 − e −x 2 ) 19. y = tg sin cosx.(D S. −sin cos(cos x) cos 2 (sin cos x) ) 20. y = e x 2 cotg3x .(DS. xe c 2 cotg3x sin 2 3x (sin 6x −3x)) 21. y = e √ 1+lnx .(DS. e √ 1+lnx 2x √ 1+lnx ) 22. y = x 1 x .(DS. x 1 x −2 (1 − lnx)) 23. y = e x .(DS. x x (1 + lnx)) 8.1. D - a . o h`am 71 24. y = x sin x .(DS. x sin x cos x · lnx + x sin x−1 sin x) 25. y = (tgx) sin x .(DS. (tg x) sin x cos xlntgx + 1 cos x ) 26. y = x sin x .(DS. x sin x sin x x +lnx ·cos x ) 27. y = x x 2 .(DS. x x 2 +1 (1 + 2lnx)) 28. y = x e x .(DS. e x x e x 1 x +lnx)) 29. y = log x 7. (DS. − 1 xlnxlog 7 x ) 30. y = 1 2a ln x − a x + a .(DS. 1 x 2 − a 2 ) 31. y = sin ln|x|.(DS. cos ln|x| x ) 32. y =ln|sin x|.(D S. cotgx) 33. y =ln|x + √ x 2 +1|.(DS. 1 √ x 2 +1 ). Trong c´ac b`ai to´an sau d ˆay (34-40) t´ınh da . o h`am cu ’ a h`am y du . o . . c cho du . ´o . ida . ng tham sˆo ´ . 34. x = a cos t, a sin t, t ∈ (0,π). y xx ?(DS. − 1 a sin 3 t ) 35. x = t 3 , y = t 2 . y xx ?(DS. − 2 9t 4 ) 36. x =1+e at , y = at + e −at . y xx ?(DS. 2e −3at − e −2at ) 37. x = a cos 3 t, y = a sin 3 t. y xx ?(DS. 1 3a sin t cos 4 t ) 38. x = e t cos t, y = e t sin t. y xx ?(DS. 2 e t (cos t −sin t) 3 ) 39. x = t − sin t, y =1−cos t. y xx ?(DS. − 1 4 sin 4 t 2 ) 40. x = t 2 +2t, y = ln(1 + t). y xx ?(DS. −1 4(1 + t) 4 ). 72 Chu . o . ng 8. Ph´ep t´ınh vi phˆan h`am mˆo . tbiˆe ´ n Trong c´ac b`ai to´an sau dˆay (41-47) t´ınh da . o h`am y ho˘a . c y cu ’ a h`am ˆa ’ ndu . o . . c x´ac d i . nh bo . ’ i c´ac phu . o . ng tr`ınh d ˜acho 41. x + √ xy + y = a. y ?(DS. 2a − 2x − y x +2y −a ) 42. arctg y x =ln x 2 + y 2 . y ?(DS. x + y x − y ) 43. e x sin y − e −y cos x =0. y ?(DS. − e x sin y + e −y sin x e x cos y + e −y cos x ) 44. x 2 y + arctg y x =0. y ?(DS. −2x 3 y −2xy 3 + y x 4 + x 2 y 2 + x ) 45. e x − e y = y −x. y ?(DS. (e y − e x )(e x+y − 1) (e y +1) 3 ) 46. x + y = e x−y . y ?(DS. 4(x + y) (x + y +1) 3 ) 47. y = x + arctgy. y ?(DS. −(2y 2 +2) y 5 ). Trong c´ac b`ai to´an sau d ˆay (48-52) t´ınh da . o h`am cu ’ a h`am ngu . o . . c v´o . ih`amd ˜a cho. 48. y = x + x 3 , x ∈ R. x y ?(DS. x y = 1 1+3x 2 ) 49. y = x +lnx, x>0. x y ?(DS. x y = x x +1 , y>0) 50. y = x + e x . x y ?(DS. x y = 1 1+y −x , y ∈ R) 51. y =chx, x>0. x y ?(DS. x y = 1 y 2 − 1 ) 52. y = x 2 1+x 2 , x<0. x y ?(DS. x y = x 3 2y 2 , y ∈ (0, 1)). 53. V´o . i gi´a tri . n`ao cu ’ a a v`a b th`ı h`am f(x)= x 3 nˆe ´ u x x 0 , ax + b nˆe ´ u x>x 0 [...]... x0 , e f (x) = ax2 + b ´ nˆu x > x0 e ’ ’ kha vi tai diˆm x = x0 (x0 = 0) ? e f (x0 − 0) x0 (DS a = , b = f (x0 ) − f (x0 − 0)) 2x0 2 ´ e Trong c´c b`i to´n (56 -62) t´ dao h`m y nˆu a a a ınh a 2 56 y = e−x 57 y = tgx √ 58 y = 1 + x2 x 59 y = arcsin 2 1 60 y = arctg x 2 (DS 2e−x (2x2 − 1)) 2 sin x ) (DS cos3 x 1 (DS ) (1 + x2)3/2 x (DS ) (4 − x2)3/2 2x (DS ) (1 + x2 )2 ´ Chu.o.ng 8 Ph´p t´ vi... ’ ´ ’ a e a a e a 54 X´c dinh α v` β dˆ c´c h`m sau: a) liˆn tuc kh˘p no.i; b) kha vi a i nˆu ´ ´ kh˘p no e a αx + β nˆu x 1 ´ e 1) f (x) = x2 ´ nˆu x > 1 e 2) α + βx2 f (x) = 1 |x| ´ nˆu |x| < 1, e ´ nˆu |x| e 1 (DS 1) a) α + β = 1, b) α = 2, β = −1; 2) a) α + β = 1, b) 3 1 α = , β = − ) 2 2 h`m y = f (x) x´c dinh trˆn tia (−∞, x0) v` kha vi bˆn ’ ’ ’ e a e a 55 Gia su a i gi´ tri... (DS 2n−1 cos 2x + n · 2 (DS 4n n!) 75 y = (4x + 1)n an ) (ax + b)n nπ ) (DS 4n−1 cos 4x + 77 y = sin4 x + cos4 x 2 3 1 ` ’ ˜ Chı dˆ n Ch´.ng minh r˘ng sin4 x + cos4 x = + cos 4x a u a 4 4 n nπ 3 π 3 78 y = sin3 x − sin 3x + n · ) (DS sin x + 4 2 4 2 ’ ˜ Chı dˆ n D`ng cˆng th´.c sin 3x = 3 sin x − 4 sin3 x a u o u 76 y = ln(ax + b) (DS (−1)n−1 (n − 1)! 8.2 Vi phˆn a 75 79 y = sin αx sin βx 1 1 π π (DS... d(uv) = udv + vdu, u vdu − udv d = , v v2 v = 0 (8 .5) 8.2 Vi phˆn a 77 ´ a ’ a 2+ Cˆng th´.c vi phˆn dy = f (x)dx luˆn luˆn thoa m˜n bˆt luˆn o u o o a a ´ ´ ´ ’ a a e o a a ınh a a x l` biˆn dˆc lˆp hay l` h`m cua biˆn dˆc lˆp kh´c T´ chˆt n`y a e o a ´ ´ e ´ ’ a e ` a a du.o.c goi l` t´nh bˆt biˆn vˆ dang cua vi phˆn cˆp 1 a ı 8.2.2 ´ Vi phˆn cˆp cao a a ´ ’ ’ ’ Gia su x l` biˆn dˆc lˆp v` h`m... arcsinx 62 y = f (ex ) (DS exf (ex ) + e2xf (ex )) ´ ´ ’ Trong c´c b`i to´n (63-69) t´nh dao h`m cˆp 3 cua y nˆu: a a a ı a e a 4(3x − 4) x (DS ) 63 y = arctg 2 (4 + x2 )3 64 y = xe−x (DS e−x (3 − x)) 65 y = ex cos x (DS −2ex (cos x + sin x)) 66 y = x2 sin x (DS −2ex (cos x + sin x)) 67 y = x32x (DS 2x (x3ln3 2 + 9x2 ln2 x + 18xln2 + 6)) 68 y = x2 sin 2x 69 y = (f (x2 ) (DS −4(2x2 cos 2x + 6x sin 2x . 0)). Trong c´ac b`ai to´an (56 -62) t´ınh da . o h`am y nˆe ´ u 56 . y = e −x 2 .(DS. 2e −x 2 (2x 2 − 1)) 57 . y =tgx.(DS. 2 sin x cos 3 x ) 58 . y = √ 1+x 2 .(DS. 1 (1 + x 2 ) 3/2 ) 59 . y = arcsin x 2 .(DS. x (4. f(x)nˆe ´ u: 1. y = 4 √ x 3 + 5 x 2 − 3 x 3 + 2. (DS. 3 4 4 √ x − 10 x 3 + 9 x 4 ) 2. y = log 2 x + 3log 3 x.(DS. ln24 xln2 · ln3 ) 3. y =5 x +6 x + 1 7 x .(DS. 5 x ln5 + 6 x ln6 − 7 −x ln7) 4 e 1 lnx .(DS. −e 1 lnx xln 2 x ) 14. y =ln e 2x + √ e 4x +1. (DS. 2e 2x √ e 4x +1 ) 15. y =ln e 4x e 4x +1 .(DS. 2 e 4x +1 ) 16. y = log 5 cos 7x.(DS. − 7tg7x ln5 ) 17. y = log 7 cos √ 1+x.(DS. − tg √ 1+x 2 √ 1+xln7 ) 18.