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Решения к Сборнику заданий по высшей математике Кузнецова Л.А. - 2. Дифференцирование. Зад.7

Задача 7. Найти производную.

7.1.

√x + √x ­

y'= ln(√x+√(x+a)) + 2√x 2√(x+a) _ 1 =

2√x √x+√(x+a) 2√(x+a)

= ln(√x+√(x+a)) + √x .

2√x 2(√x+√(x+a))√(x+a)

7.2.

y'= 1+x/√(a2+x2) = x+√(a2+x2) = 1 .

x+√(a2+x2) (x+√(a2+x2))√(a2+x2) √(a2+x2)

7.3.

y'= 1 _ 2/√x = 2+√x-2 = 1 .

√x 2+√x √x(2+√x) 2+√x

7.4.

y'= √(1-ax4) * 2x√(1-ax4)+2ax5/√(1-ax4) = 2√(1-ax4)+2ax4

x2 1-ax4 x-ax5

7.5.

1 + 1 _

y'= 2√x 2√(x+1) = √(x+1)+√x = 1 .

√x+√(x+1) 2√(x2+x)( √x+√(x+1)) 2√(x2+x)

7.6.

y'= a2-x2 * 2x(a2-x2)+2x(a2+x2) = 4xa2

a2+x2 (a2-x2)2 a4-x4

7.7.

y'= 2ln(x+cosx)* 1-sinx .

x+cosx

7.8.

y'= -3ln2(1+cosx)* -sinx .

1+cosx

7.9.

y'= 1-x2 * 2x(1-x2)+2x3 = 2 .

x2 (1-x2)2 x(1-x2)

7.10.

y'= ctg(π/4+x/2) = 2 = 2 .

2cos2(π/4+x/2) sin(π/2+x) cosx

7.11.

y'= 1-2x * 2(1-2x)+2(1+2x) = 1 .

4+8x (1-2x)2 2-8x2

7.12.

_

y'= 1+ (x+√2)(x+√2-x+√2) = 1+ 1 .

(x-√2)(x+√2)2 x2-2

7.13.

y'= cos((2x+4)/(x+1)) * 2x+2-2x-4 = -2ctg((2x+4)/(x+1))

sin((2x+4)/(x+1)) (x+1)2 (x+1)2

7.14.

y'= 1 * 1 * 1 = 1 = lntgx _

ln16*log5tgx tgx*ln5 cos2x ln4*ln5*sin2x*log5tgx 2sin2x*ln32

7.15.

y'= 1 = lntgx .

4ln22*cos2x*tgx*log2tgx 2sin2x*ln32

7.16.

y'= 1/2*(coslnx+sinlnx+x(-1/x*sinlnx+1/x*coslnx))= coslnx

7.17.

y'= -sin((2x+3)/(x+1))*2x+2-2x-3 = ctg((2x+3)/(x+1))

cos((2x+3)/(x+1)) (x+1)2 (x+1)2

7.18.

y'= -lge = -2lge .

lnctgx*ctgx*sin2x lnctgx*sin2x

7.19.

y'= 4x3 = 2x3 .

2(1-x4)lna lna(1-x4)

7.20.

1 * 4tgx _

y'= cos2x 2√2cos2x√1+2tg2x = 2tgx _

√2tgx+√(1+2tg2x) cos4x√(1+2tg2x)( √2tgx+√(1+2tg2x))

7.21.

y'= 1 * 1 * -2e2x = -ex_

arcsin√(1-e2x) √(1-1+e2x) 2√(1-e2x) √(1-e2x)arcsin√(1-e2x)

7.22.

y'= 1 * 1 * -4e4x = -2e2x_

arccos√(1-e4x) √(1-1+e4x) 2√(1-e4x) √(1-e4x)arccos√(1-e4x)

7.23.

y'= b+b2x/√(a2+b2x2) = b _

bx+√(a2+b2x2) √(a2+b2x2)

7.24.

y'= √(x2+1)-x√2 * (x/√(x2+1)+√2)( √(x2+1)-x√2)-(x/√(x2+1)-√2)( √(x2+1)+x√2)=

√(x2+1)+x√2 (√(x2+1)-x√2)2

= (x+√(x2+1))(√(x2+1)-x√2)-(x-√2√(x2+1))(√(x2+1)+x√2) =

√(x2+1)(√(x2+1)-x√2)2

= 2√2 _

√(x2+1)(√(x2+1)-x√2)2

7.25.

y'= -1/(2√x3) = -1 _

arcos(1/√x) 2√x3arccos(1/√x)

7.26.

y'= ex+e2x/√(1+e2x) = ex _

ex+√(1+e2x) √(1+e2x)

7.27.

√5-tg(x/2)+√5+tg(x/2)

y'= √5-tg(x/2) * 2cos2(x/2) 2cos2(x/2) = √5 _

√5+tg(x/2) (√5-tg(x/2))2 (5-tg2(x/2))cos2(x/2)

7.28.

sin(1/x)+lnxcos(1/x)

y'= sin(1/x)* x x2 = 1 + ctg(1/x)

lnx sin2(1/x) xlnx x2

7.29.

y'= cos(1+1/x) * -1/x2 = -ctg(1+1/x) _

lnsin(1+1/x) sin(1+1/x) x2lnsin(1+1/x)

7.30.

y'= 3ln2ln2x*3ln2x*1 = 6 _

ln3ln3x ln3x x xlnln2xlnx

7.31.

y'= 2lnln3x*3ln2x* 1 = 6 _

ln2ln3x ln3x x xlnln3xlnx