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Chapter 3 66

# Chapter 3 66 - 216 CHAPTER 3 DIFFERENTIATION RULES...

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Unformatted text preview: 216 CHAPTER 3 DIFFERENTIATION RULES sinh(lnac) # (61” — e_lm)/2 _ a: — (61”)_1 w — w‘l cosh(lnx) _ (elnm+e—1nm)/2 _ m+(elnz)_1 m+\$—1 \$e1/x:(m2—1)/x:x2—l :c-I—l/a: (x2+1)/ac m2+1 17. tanh(1n m) : NIH 1 + tanhx 1 + (sinh m)/coshac coshac + sinhm (67” -I- 6—1)+%(61 — 6‘1) 18. —— : __———— : ____ z 11 tanhm 1— (sinh cc) /cosh:v coshx — sinh cc ﬂex + 6”) ! ﬂex — e—E) em + 6—1 + e3 — 6"” 2e“ 21 : _————— : : 6 6m + 6"” — 6m + 6“ 26": coshm + sinhm e1 22 0r: Using the results of Exercises 9 and 10, ———,—- : cosh :c — s1nh :3 6—1 19. By Exercise 9, (coshm + sinh cc)" : (e\$)" : em = cosh ms + sinh nae. 20.sinhx=§ :> cschmzl sinhm=\$ cosh2w:sinh2m+1:3+1:3§ => coshm=§ since 4 3 16 16 4 cosh x > 0). secha: = 1/coshx : §,tanh a: : sinhm/coshm 2 g/L: = ,and cothm = 1/tanhm = 3 mice 21. tanha;=§>0,socc>0. cothmzl/tanhm:%,sech2\$:1#tanh2m=1—( )22593 => cschzc = 1/sinha: : 3. _ _ 1 y _ CSChX _ sinhx y : SCChx : (:05th 1. h 1. ex—e‘w {I 1. 1—e*% 1—0 1 : Pé . :. m ——— z —— z 23' (a) 11—>nc}otan :13 gal—{go e“ + e‘\$ e—3E mime 1 + (3—29” 1 + 0 em—e’I ”5 ezm—l 0—1 (b) \$113100 tanhx \$112100 63; + 67m ea: EJIPOO 621 + 1 0 + 1 (B __ e—(E (0) lim sinhn: : lim _e___ : 00 Id“) I—NDO 2 (d) lim sinha: : lim Lie— z —oo maioo \$—+—OO 2 2 (6) 11320 SBChII) 93121010 6‘” + 6‘1 6“” + 6—3” 671‘ _ 1 + 6-2\$ 1 + 0 - _' . = ———:———:1 0:Use arta (f) mascot“ - 3320 ex # 3:“; 1 _ 1 _ 0 I r P < >1 (g) 1'1m+ cotha: = lim+ 2:2: : 00, since sinh :13 —> 0 through positive values and cosha: —> 1. maO zaO (h) lim cothzv : lim :33: z —00, since sinh :c —> 0 through negative values and coshac ——> 1, 190’ z—>0' 2 (i) lim cschrc : lim —— : 0 :z—v—oo maroo em — egm ...
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