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Chapter 3 29 - 12 13 14 15 16 17 18 19 20 21 23 24 25...

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Unformatted text preview: 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 23. 24. 25. SECTION 3.5 THE CHAIN RULE D sec2 t f(t) : \3/1+tant : (1 +tant)l/3 : f’(t) : %(1+tant)“2/35ec2t = —-——— 3 ‘3/(1+ta.nt)2 y = cos(a3 + m3) i y/ : — sin(a3 + :03) ~3$2 [a3 isjust a constant] 2 +3222 sin(a3 + 933) y = a3 + cos3 a: :> y' : 3(cos $)2(— sinm) [a3 is just a constant] 2 —3sinat:cos2 m d y = 6—7” 2 y’ = ($me (—mm) = e‘m” (—m) = “me—m1 y : 4sec 53v => 3/ = 4360 5w tan 5$(5) : 20 sec 590 tan 5w 9(33) = (1 + 4m)5(3 + :8 — $2)8 => g’(:c) : (1+ 435)5 - 8(3 + :1: — x2)7(1— 2x) + (3 + a: — x2)8 ~ 5(1+ 4x)4 - 4 : 4(1 + 4$)4(3 + :1: — x2)7 [2(1 + 42:)(1 — 2x) + 5(3 + m — 3.3)] = 4(1 + 4$)4(3 —— an — m2)7 [(2 + 4x — 163:2) + (15 + 517 — 5%)] = 4(1 + 4:0)4(3 —— m — m2)7 (17 + 93: — 21x2) W) = (754 ~ 1)3(t3 +1)4l is h’(t) = (t4 — 1)3 -4(t3 + 1)3(3t2) + (t3 +1)4.3(14 — 1)2(4t3) : 12t2(t4 — 1)2(t3 +1)3 [(51 21) + t(t3 + 1)] :12t2(t4 —1)2(t3 +1)3 (22:4 + t — 1) y : (2:1: — 5)4(8x2 — 5) ‘3 :> y’ : 4(23: — 5)3(2)(8:i2 5)‘3 : (2m 5)4( 3)(83:2 5)‘4 (16$) : 8(2x — 5)3(8$2 ~ 5)‘3 — 4833(290 — 5)4(8m2 — 5) ‘4 [This simplifies to 8(2x ~ 5)3(8352 — 5) ‘4(—4a:2 + 301' — 5).] y : (222 +1)($2 + 2)”3 9' : 293W + 2W3 + ($2 + 1) (g) (x2 + 2)‘2/3 (2x) : 2m(:c2 + 2)1/3[1 + 7562232)] 2 _z 2 y : :ce => 14' = me‘m2(—2x) + e“ -1: 6W2 (—2er + 1) 2 8—12(1 — 23:2) . y : 675$ cos 3m 2?, y’ : e751 (—3sin 3x) + (cos 3$)(—5€75$) = —e‘51(3sin 3:0 + 5cos 3:15) d ZCOSE y = e :> y' = e“°” - % (mcosm) = em“Lr [m(—sinm) + (00521:) - 1] = emosflcosm —;L‘sina:) Using Formula 5 and the Chain Rule, 3; : 10H2 => y' : 101-12(1n10) - i (1 ~ $2) : —2x(in 10)10HZ. 2+1 2—1 1/2 F : : (Z) 2+1 (2+1) :> —1/2 F'(z) :64) .1 2—1 -1 2+11/2_<z+1)<1>—<z—1)<1) 2 2+1 d2 2+1 2 2+1 (2+1)2 _1(2+1)1/272+1—2+1H1(2+1)1/2 2 1 2(24)“2 (2+1)2 _ 2(2—1)1/2.(Z+1)2 _ (2+1)1/2(2+1)3/2 179 ...
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