Реферати, курсові, дисертації, дипломи

[ Рішення до Збірника завдань з вищої математики Кузнєцова Л.А. - 2. Диференціювання. Зад.6 ]

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	 Задача 6. Знайти похідну. 6.1. <IMG SRC=ref-0_747527979-15266.coolpic NAME=Object1 ALIGN=ABSMIDDLE WIDTH=285 HEIGHT=48> e x + 2 e 2 x + e x y '= 1 - √ (e 2 x + e x +1) = 2 + e x + √ (e 2 x + e x +1) - e x √ (e 2 x + e x +1) -2 e 2 x - e x = 2 + e x +2 √ (e 2x + e x +1) 2 + e x +2 √ (e 2x + e x +1) = (2-e x) √ (e 2x + e x +1) +2 + e x-2e x 2 + e x +2 √ (e 2x + e x +1) 6.2. <IMG SRC=ref-0_747543245-17293.coolpic NAME=Object2 ALIGN=ABSMIDDLE WIDTH=259 HEIGHT=33> y '= 1 / 4 * e 2x (2-sin2x-cos2x) +1 / 8 * e 2x (-2cos2x +2 sin2x) = 1 / 8 * e 2x (4-2sin2x-2cos2x-2cos2x +2 sin2x) = 1 / 8 * e 2x (4-4cos2x) = e 2x * sin2x 6.3. <IMG SRC=ref-0_747560538-23275.coolpic NAME=Object3 ALIGN=ABSMIDDLE WIDTH=152 HEIGHT=56> y '= 1 * 1 * 2e x = E x. 2 Січень + (e x -3) 2 4 e 2x-6e x +10 4 6.4. <IMG SRC=ref-0_747583813-22014.coolpic NAME=Object4 ALIGN=ABSMIDDLE WIDTH=148 HEIGHT=56> y '= 1 * 1-2 x * -2 x ln2 (1 +2 x) - (1-2 x) 2 x ln2 = (2 x -1) 2 x ln4 = 2 x (2 x -1) ln4 1 +2 x (1 +2 x) 2 ln4 (1 +2 x) 3 (1 +2 x) 3 6.5. <IMG SRC=ref-0_747605827-19584.coolpic NAME=Object5 ALIGN=ABSMIDDLE WIDTH=239 HEIGHT=67> e x (√ (e x +1) +1) _ e x (√ (e x +1) -1) y '= e x + √ (e x +1) +1 * 2 √ (e x +1) 2 √ (e x +1) = √ (e x +1) √ (e x +1) -1 (√ (e x +1) +1) 2 = E x + E x √ (e x +1) + e x-e x √ (e x +1) + e x = √ (e x +1) √ (e x +1) 2e x √ (e x +1) 6.6. <IMG SRC=ref-0_747625411-25937.coolpic NAME=Object6 ALIGN=ABSMIDDLE WIDTH=161 HEIGHT=53> y "= 2 / 3 * 3 / 2 * √ (arctge x) * e x = E x √ (arctge x) 1 + e x 1 + e x 6.7. <IMG SRC=ref-0_747651348-17760.coolpic NAME=Object7 ALIGN=ABSMIDDLE WIDTH=241 HEIGHT=53> y '= 2e x - 2e x = E x 2 (e 2x +1) 1 + e 2x 1 + e 2x 6.8. <IMG SRC=ref-0_747669108-22625.coolpic NAME=Object8 ALIGN=ABSMIDDLE WIDTH=288 HEIGHT=72> 6.9. <IMG SRC=ref-0_747691733-19373.coolpic NAME=Object9 ALIGN=ABSMIDDLE WIDTH=264 HEIGHT=72> y "= 2 / ln 2 * ((2 x ln 2) / (2 √ (2 x -1)) - (2 x ln 2) / (1 +2 x -1)) = 2 x / √ (2 x -1) -2 6.10. <IMG SRC=ref-0_747711106-17696.coolpic NAME=Object10 ALIGN=ABSMIDDLE WIDTH=319 HEIGHT=67> e x (√ (1 + e x) +1) _ e x (√ (1 + e x) -1) y '= 2 √ (1 + e x) + 2 e x (x -2) _ √ (1 + e x) +1 * 2 √ (1 + e x) 2 √ (1 + e x) = 2 √ (1 + e x) √ (1 + e x) -1 (√ (1 + e x) +1) = Xe x +2 _ 2e x √ (1 + e x) +2 e x = Xe x. √ (1 + e x) e x √ (1 + e x) ( √ (1 + e x) +1) √ (1 + e x) 6.11. <IMG SRC=ref-0_747728802-27268.coolpic NAME=Object11 ALIGN=ABSMIDDLE WIDTH=255 HEIGHT=63> y '= αe αx (αsinβx-βcosβx) + e αx (αβcosβx + β 2 sinβx) = α 2 + β 2 = E αx (α 2 sinβx + β 2 sinβx) = e αx sinβx α 2 + β 2 6.12. <IMG SRC=ref-0_747756070-27308.coolpic NAME=Object12 ALIGN=ABSMIDDLE WIDTH=255 HEIGHT=63> y '= αe αx (βsinβx-αcosβx) + e αx (β 2 cosβx + αβsinβx) = α 2 + β 2 = E αx (β 2 cosβx +2 αβsinβx-α 2 cosβx) α 2 + β 2 6.13. <IMG SRC=ref-0_747783378-28802.coolpic NAME=Object13 ALIGN=ABSMIDDLE WIDTH=315 HEIGHT=77> y '= ae ax * ┌ 1 + acos2bx +2 bsin2bx ┐ + e ax ┌-2absin2bx +4 b 2 cos2bx ┐ = └ 2a 2 (a 2 +4 b 2) ┘ └ 2 (a 2 +4 b 2) ┘ = E ax / 2 * (1 + cos2bx) = e ax cos 2 bx 6.14. <IMG SRC=ref-0_747812180-16501.coolpic NAME=Object14 ALIGN=ABSMIDDLE WIDTH=221 HEIGHT=53> y '= 1 - e x - E x = 1 - e x-e x-e 2x = 1 + e 2x. (1 + e x) 2 1 + e x (1 + e x) 2 (1 + e x) 2 6.15. <IMG SRC=ref-0_747828681-17083.coolpic NAME=Object15 ALIGN=ABSMIDDLE WIDTH=372 HEIGHT=45> 3 / 6 * e x / 6 * √ (1 + e x / 3) + 1 / 3 * e x / 3 (1 + e x / 6) y '= 1 - 2 √ (1 + e x / 3) _ 3 / 6 * e x / 6 = (1 + e x / 6) √ (1 + e x / 3) 1 + e x / 3 = 1 - e x / 6 + e x / 2 + e x / 3 + e x / 2 _ E x / 6 = 1 - e x / 3-e x / 6. 2 (1 + e x / 6) (1 + e x / 3) 2 (1 + e x / 3) 2 (1 + e x / 6) (1 + e x / 3) 6.16. <IMG SRC=ref-0_747845764-20562.coolpic NAME=Object16 ALIGN=ABSMIDDLE WIDTH=133 HEIGHT=53> y '= 1 - 8e x / 4 = 1 - 2e x / 4 . 4 (1 + e x / 4) 2 (1 + e x / 4) 2 6.17. <IMG SRC=ref-0_747866326-16389.coolpic NAME=Object17 ALIGN=ABSMIDDLE WIDTH=283 HEIGHT=48> e x + e 2x y '= √ (e 2x -1) _ e-x = E x (e x + √ (e 2x -1)) _ e-x * e x = e x -1. e x + √ (e 2x -1) √ (1-e-2x) (e x + √ (e 2x -1)) √ (e 2x -1) √ (e 2x -1) √ (e 2x -1 ) 6.18. <IMG SRC=ref-0_747882715-15041.coolpic NAME=Object18 ALIGN=ABSMIDDLE WIDTH=323 HEIGHT=48> e 2x y '= 1 + e-x arcsine x - e-x * e x + √ (1-e 2x) = √ (1-e 2x) 1 + √ (1-e 2x) = 1 + e-x arcsine x - 1 + e 2x = √ (1-e 2x) (1 + √ (1-e 2x)) √ (1-e 2x) = E-x arcsine x 6.19. <IMG SRC=ref-0_747897756-14492.coolpic NAME=Object19 ALIGN=ABSMIDDLE WIDTH=416 HEIGHT=41> y '= 1 - e x + E-x / 2 arctge x / 2 - e-x / 2 * e x / 2 _ e x / 2 arctge x / 2 = 1 + e x 1 + e x 1 + e x = 1 - e x + 1 + arctge x / 2 * 1-e x = Arctge x / 2 * 1-e x. 1 + e x 1 + e x e x / 2 (1 + e x) e x / 2 (1 + e x) 6.20. <IMG SRC=ref-0_747912248-27409.coolpic NAME=Object20 ALIGN=ABSMIDDLE WIDTH=92 HEIGHT=60> y '= 3x 2 e x3 (1 + x 3)-3e x3 x 2 = 3x 5 e x3 (1 + x 3) 2 (1 + x 3) 2 6.21. <IMG SRC=ref-0_747939657-27654.coolpic NAME=Object21 ALIGN=ABSMIDDLE WIDTH=237 HEIGHT=69> y '= b * me mx √ a = e mx. m √ (ab) (b + ae 2mx) √ b b + ae 2mx 6.22. <IMG SRC=ref-0_747967311-19743.coolpic NAME=Object22 ALIGN=ABSMIDDLE WIDTH=224 HEIGHT=48> y '= e 3 ^ √ x / 3 √ x (3 √ x 2 -2 3 √ x +2) +3 e 3 ^ √ x (2 / (3 березня √ x) -2 / (3 березня √ x 2) ) = e 3 ^ √ x 3 ^ √ x = кубічний корінь з х 6.23. <IMG SRC=ref-0_747987054-22982.coolpic NAME=Object23 ALIGN=ABSMIDDLE WIDTH=236 HEIGHT=67> (E x +2 e 2x _ E x) (√ (1 + e x + e 2x)-e x +1) _ (e x +2 e 2x _ E x) (√ (1 + e x + e 2x)-e x -1) y '= √ (1 + e x + e 2x)-e x +1 * 2 √ (1 + e x + e 2x) 2 √ (1 + e x + e 2x) = √ (1 + e x + e 2x)-e x -1 (√ (1 + e x + e 2x)-e x +1) 2 = E x (1 +2 e 2x -2 √ (1 + e x + e 2x)) = 1. (E x (1 +2 e 2x -2 √ (1 + e x + e 2x))) √ (1 + e x + e 2x) √ (1 + e x + e 2x) 6.24. <IMG SRC=ref-0_748010036-25465.coolpic NAME=Object24 ALIGN=ABSMIDDLE WIDTH=176 HEIGHT=59> y '= cosxe sinx (x-1/cosx) + e sinx (1-sinx/cos 2 x) = e sinx (xcosx-sinx/cos 2 x) 6.25. <IMG SRC=ref-0_748035501-17941.coolpic NAME=Object25 ALIGN=ABSMIDDLE WIDTH=316 HEIGHT=56> y '= e x / 2 ((x 2 -1) cosx + (x-1) 2 sinx) + e x / 2 (2xcosx-(x 2 -1) sinx +2 (x-1) sinx + (x-1 ) 2 cosx) = = E x / 2 (x-1) (5x +3) cosx 6.26. <IMG SRC=ref-0_748053442-17920.coolpic NAME=Object26 ALIGN=ABSMIDDLE WIDTH=169 HEIGHT=37> y '= e x + e-x = e 3x + e x. 1 + (e x-e-x) 2 e 4x-e 2x +1 6.27. <IMG SRC=ref-0_748071362-16181.coolpic NAME=Object27 ALIGN=ABSMIDDLE WIDTH=456 HEIGHT=48> y '= e 3 ^ √ x / 3 √ x 2 (3 √ x 5 -5 3 √ x 4 +20 x-60 3 √ x 2 +120 3 √ x-120) + e 3 ^ √ x (5 березня √ x 2 -20 3 √ x +20-120 / 3 √ x +120 / 3 √ x 2) = e 3 ^ √ x (x-40) 6.28. <IMG SRC=ref-0_748087543-26212.coolpic NAME=Object28 ALIGN=ABSMIDDLE WIDTH=113 HEIGHT=56> y '=-3e 3x sh 3 x +3 e 3x sh 2 xchx = e 3x (chx-shx) 3sh 6 x sh 4 x 6.29. <IMG SRC=ref-0_748113755-15477.coolpic NAME=Object29 ALIGN=ABSMIDDLE WIDTH=208 HEIGHT=35> y '=-e-x + e 2x = √ (e 4x-e 2x) - √ (e-2x -1) = √ (e 2x -1) - √ (1-e 2x) √ (1-e-2x) √ (1-e 2x) √ (1-e-2x) * √ (1-e 2x) √ (1-e-2x) √ (1-e 2x) 6.30. <IMG SRC=ref-0_748129232-15522.coolpic NAME=Object30 ALIGN=ABSMIDDLE WIDTH=227 HEIGHT=53> y '= xe-x2 (x 4 +2 x 2 +2) -1 / 2 * e-x2 (4x 3 +4 x) = x 5 e-x2 6.31. <IMG SRC=ref-0_748144754-25229.coolpic NAME=Object31 ALIGN=ABSMIDDLE WIDTH=93 HEIGHT=60> y '= 2xe x2 (1 + x 2)-2e x2 x = 2x 3 e x2 (1 + x 2) 2 (1 + x 2) 2 
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