# Реферат: 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*log5 tgx tgx*ln5 cos2 x ln4*ln5*sin2x*log5 tgx 2sin2x*ln3 2

7.15.

y'= 1 = lntgx .

4ln2 2*cos2 x*tgx*log2 tgx 2sin2x*ln3 2

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*sin2 x lnctgx*sin2x

7.19.

y'= 4x3 = 2x3 .

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

7.20.

1 * 4tgx _

y'= cos2 x 2√2cos2 x√1+2tg2 x = 2tgx _

√2tgx+√(1+2tg2 x) cos4 x√(1+2tg2 x)( √2tgx+√(1+2tg2 x))

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+b2 x/√(a2 +b2 x2 ) = b _

bx+√(a2 +b2 x2 ) √(a2 +b2 x2 )

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√x3 arccos(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) x2 lnsin(1+1/x)

7.30.

y'= 3ln2 ln2 x *3ln2 x *1 = 6 _

ln3 ln3 x ln3 x x xlnln2 xlnx

7.31.

y'= 2lnln3 x *3ln2 x * 1 = 6 _

ln2 ln3 x ln3 x x xlnln3 xlnx

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