Главная Учебники - Разные Лекции (разные) - часть 26
Задача 11
. Найти производную. 11.1. lny= 1/2*ln2
arctgx y'= (arctgx)1/2*ln(arctgx)
(lnarctgx)/(arctgx*(1+x2
)) 11.2. lny= ln2
sin√x y'= ((sin√x)lnsin√x
*ctg√x*lnsin√x)/√x 11.3. lny= 5ex
lnsinx y'= (sinx)5e^x
(5ex
lnsinx+5ex
ctgx) 11.4. lny= ex
lnarcsinx y'= (arcsinx)e^x
(ex
lnarcsinx+ex
/√(1-x2
)) 11.5. lny= 3x
lnlnx y'= (lnx)3^x
(3x
ln3lnlnx+3x
/(xlnx)) 11.6. lny= arcsinxlnx y'= xarcsinx
(lnx/√(1-x2
)+arcsinx/x) 11.7. lny= 2ex
lnctg3x y'= 2(ctg3x)2e^x
(ex
lnctg3x-3ex
/(ctg3x*sin2
3x)) 11.8. lny= etgx
lnx y'= xe^tgx
((etgx
lnx)/cos2
x+etgx
/x) 11.9. lny= 4exlntgx y'= (tgx)4e^x
(4ex
lntgx+ 4ex
) = (tgx)4e^x
(4ex
lntgx+ 4ex
) = tgxcos2
x sinxcosx 11.10. lny=exlncos5x y'= (cos5x)e^x
(ex
lnco5x-5ex
sin5x
)= ex
(cos5x)e^x
(lncos5x-5tg5x) cos5x 11.11. lny= 8ln2
(xsinx) y'= 16(xsinx)8ln(xsinx)
ln(xsinx)(sinx+xcosx) 11.12. lny= chxln(x-5) y'= (x-5)chx
(shxln(x-5)+chx/(x-5)) 11.13. lny= tgxln(x3+4) y'= (x3
+4)tgx
(ln(x3
+4)/cos2
x+(3x2
tgx)/(x3
+4)) 11.14. lny= sinx3
lnx y'= xsinx^3
(3x2
cosx3
lnx+(sinx3
)/x) 11.15. lny= shxln(x2
-1) y'= (x2
-1)shx
(chxln(x2
-1)+(2xshx)/(x2
-1)) 11.16. lny= ctgxln(x4+5) y'= (x4
+5)ctgx
(-(ln(x4
+5))/sin2
x+(4x3
ctgx)/(x4
+5)) 11.17. lny= 5x/2*lnsinx y'= (sinx)5x/2
(2,5lnsinx+(5xcosx)/sinx)= (sinx)5x/2
(2,5lnsinx+5xctgx) 11.18. lny= cosxln(x2+1) y'= (x2
+1)cosx
(-sinxln(x2
+1)+(2xcosx)/(x2
+1)) 11.19. lny= x19
ln19+19lnx y'= 19x^19
x19
(19x18
ln19+19/x) 11.20. lny= 3x
lnx+xln2 y'= x3^x
2x
(3x
ln3lnx+3x
/x+ln2) 11.21. lny= e1/x
lnsin√x y'= (sin√x)e^1/x
(-(e1/x
lnsin√x)/x2
+(e1/x
cos√x)/(2√xsin√x))= = e1/x
(sin√x)e^1/x
((ctg√x)/(2√x)-(lnsin√x)/x2
) 11.22. lny= ectgx
lnx y'= xe^ctgx
(-(ectgx
lnx)/sin2
x+ectgx
/x)= ectgx
xe^ctgx
(1/x-(lnx)/sin2
x) 11.23. lny= ecosx
lnx y'= xe^cosx
(-ecosx
sinxlnx+ecosx
/x)= ecosx
xe^cosx
(1/x-sinxlnx) 11.24. 11.25. lny= esinx
lnx y'= xe^sinx
(esinx
cosxlnx+esinx
/x)= ecosx
xe^cosx
(cosxlnx+1/x) 11.26. lny= ln2
(tgx)/4 y'= (tgx)ln(tgx)/4
/*(ln(tgx))/(2cos2
x) 11.27. lny= earctgx
lnx y'= xe^arctgx
((earctgx
lnx)/(1+x2
)+ earctgx
/x) 11.28. lny= thxln(x8
+1) y'= (x8
+1)thx
((ln(x8
+1))/ch2
x+(8x7
thx)/(x8
+1)) 11.29. lny= 29x
lnx+xln29 y'= x29^x
29x
(29x
ln29lnx+29x
/x+ln29) 11.30. lny= ln2
(cos2x)/4 y'= (cos2x)ln(cos2x)/4
(-ln(cos2x)sin2x)/cos2x 11.31. lny= ex
lnx+9lnx y'= xe^x
x9
(ex
lnx+ex
/x+9/x)
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