Решения к Сборнику заданий по высшей математике Кузнецова Л.А. - 2. Дифференцирование. Зад.5
Задача 5. Найти производную.
5.1.
(9x2+8x-1)(x+1)1/2 – (3x3+4x2-x-2)
y'=2/15* ___________________2(1+x)1/2 =
1+x
= 2/15* (2x+2)(9x2+8x-1)-3x3-4x2+x+2 =
2(x+1)3/2
=2/15* 18x3+16x2-2x+18x2+16x-2-3x3-4x2+x+2 =
2(x+1)3/2
= 2/15* 15x3+30x2+15x =
2(x+1)3/2
= x(x+1)2 = x(x+1)1/2
(x+1)3/2
5.2.
3x3*4x(x2+1)1/2+x(2x2-1) -9x2(2x2-1)(x2+1)1/2
y'= (x2+1)1/2 =
9x6
= 12x4(x2+1)+3x4(2x2-1)-9x2(2x2-1)(1+x2) =
9x6(x2+1)1/2
= 12x4+12x6+6x6-3x4-18x4-18x6+9x2+9x4 =
9x6(x2+1)1/2
= 9x2 = 1 .
9x6(x2+1)1/2 x4(x2+1)1/2
5.3.
y'= (4x3-16x)(x2-4)-(x4-8x2)2x = 4x5-16x3-16x3+64x-2x5+16x3 =
2(x2-4)2 2(x2-4)2
=2x5-16x3+64x =x(x2-4)2+16x = x+ 16x2 .
2(x2-4)2 (x2-4)2 (x2-4)2
5.4.
(4x-1)√(2+4x) – 2(2x2-x-1)
y'= √(2+4x) = (4x-1)(2+4x)-4x2+x+1 =
3(2+4x) 3(2+4x)√(2+4x)
= 12x2+5x-1 .
3(2+4x)√(2+4x)
5. 5.
8x19√(1+x8)+ 4x19(1+x8) – 12x11(1+x8)3/2
y'= √(1+x8) =
12x24
= 12x19(1+x8)-12x11(1+x8)2 =
12x24√(1+x8)
= x11(x16-2x8+1) = (x8-1)2 .
x24√(1+x8) x13√(1+x8)
5.6.
2x√(1-3x4) + 6x5
y'= √(1-3x4) = 2x(1-3x4)+6x5 = x .
2(1-3x4) 2(1-3x4)√(1-3x4) √(1-3x4)3
5.7.
y= (2x(4+x2)√(4+x2)+3/2√(4+x2)*2x)x5-(x2-6)(4+x2)√(4+x2)*5x4 =
120x10
= √(4+x2)(8x6+2x8+3x6-20x6-5x8+30x6+120x4) =
120x10
= √(4+x2)(7x2-x4+40)
40x6
5.8.
y= 3/2√(x2-8)*2x4-(x2-8)√(x2-8)*18x2 =
6x6
√(x2-8)(x4-6x4+48x2) = √(x2-8)(48-5x2)
3x6 3x4
5.9.
9x3(2+x3)2/3-(4+3x3)((2+x3)2/3+2/3* 3x3)
y'= (2+x3)1/3 =
x2(2+x3)4/3
= 9x3(2+x3)-(4+3x3)(2+3x3) = 8 .
x2(2+x3)5/3 x2(2+x3)5/3
5.10.
y'= √(x)*(2(1+x3/4)*3/4x5/4-(1+x3/4)2*3/2*√(x)) =
3(1+x3/4)2/3*x6/4
= √(x)(x3/2-1)
2x(1+x3/2)2/3
5.11.
(6x5+3x2)√(1-x3) + 3x2(x6+x3-2)
y' = 2√(1-x3) =
1-x3
=(2-2x3)(6x5+3x2)+3x8+3x5-6x2 = (9x5-9x8) = 9x5 .
2(1-x3)3/2 2(1-x3)3/2 2√(1-x3)
5.12.
2x4√(4+x2)+ x4(x2-2) -3x2(x2-2)√(4+x2)
y'= √(4+x2) =
24x6
= 2x4(4+x2)+x4(x2-2)-3x2(x2-2)(4+x2) = 1
24x6 x4
5.13.
2x√(1+2x2)- 2x(1+x2)
y'= √(1+2x2) = x(1+2x2)-x(1+x2) = x3 .
2(1+2x2) (1+2x2)3/2 (1+2x2)3/2
5.14.
y'= ((3x+2)/(2√(x-1))+3√(x-1))x2-2x√(3x+2) =
4x4
= x2(3x+2)+6x2(x-1)-4x(x-1)(3x+2) = 9x3-12x2+8x = 9x2-12x+8
4x2√(x-1) 4x2√(x-1) 4x√(x-1)
5.15.
y'= 3/2*√(1+x2)*2x4-3x2(1+x2)3/2 = √(1+x2)*(x4-x2-x4) = -√(1+x2)
3x6 x6 x4
5.16.
(6x5+24x2)√(8-x3)+3x2(x6+8x3-128)
y'= 2√(8-x3) =
8-x3
= (16-2x3)(6x5+24x2)+3x2(x6+8x3-128) = 72x5-9x8 = 9x5
2(8-x3)3/2 2(8-x3)3/2 2√(8-x3)
5.17.
x2(x-2)+x2√(2x+3)-(2x2-4x)√(2x+3)
y'= √(2x+3) =
x4
= x2(x-2+2x+3)-(2x2-4x)(2x+3) = 3x2-x3+12x = 3x-x2+12
x4√(2x+3) x4√(2x+3) x3√(2x+3)
5.18.
y'=-2x5√(x3+1/x)+(1-x2)*1/5*(x3+1/x)4/5*(3x2-1/x2)=1/5*(x3+1/x)4/5(3x2-1/x2-3x4+1)-2x(x3+1/x)1/5
5.19.
4x4√(x2-3)+x4(2x2+3) - 3x2(2x2+3)√(x2-3)
y' = √(x2-3) =
9x6
= 4x4(x2-3)+x4(2x2+3)-3x2(2x2+3)(x2-3) = 27x2 = 3 .
9x6√(x2-3) 9x6√(x2-3) x4√(x2-3)
5.20.
y'= (x2+5)3/2-3/2*(x-1)√(x2+5)*2x = √(x2+5)(5+3x-2x2)
(x2+5)3 (x2+5)3
5.21.
2x2√(x2-x)+(2x-1)(2x+1)x2-2x(2x+1)√(x2-x)
y'= √(x2-x) =
x4
= x2(2x2-2x+4x2-1)-(4x2+2x)(x2-x) = 2x2+1
x4 x2
5.22.
_ 1+√x _ 1-√x
y' = √((1+√x)/(1-√x))* 2√x 2√x =
(1+√x)2
= -2√((1+√x)/(1-√x)) = -1 .
2√x(1+√x)2 √(x(1-x))(1+√x)
5.23.
√(x2+4x+5) - x(x+2)
y' = √(x2+4x+5) = - 2x2-6x-5 .
(x+2)2(x2+4x+5) (x+2)2(x2+4x+5)3/2
5.24.
2x+1 -3(x2+x+1)1/3
y' = (x2+x+1)2/3 = -3x2-x-2 .
(x+1)2 (x+1)2(x2+x+1)2/3
5.25.
y'= 3√((x-1)4/(x+1)2)*(x-1)2-2(x-1)(x+1) = -3√((x-1)4/(x+1)2)*x2+2x-3 =
(x-1)4 (x-1)4
= 3-x2-2x
(x2-1)2/3(x-1)2
5.26.
√(x2+2x+7)-(x+1)(x-1)
y' = √(x2+2x+7) = x2+2x+7-x2-8x-7 = -x .
6(x2+2x+7) 6(x2+2x+7)3/2 (x2+2x+7)3/2
5.27.
y' = (x2+x+1)(√(x+1)+x/(2√(x+1)))-(2x2+x)√(x+1) =
(x2+x+1)2
= (3x+2)(x2+x+1)-(4x2+2x)(x+1) = -x3-x2+3x+2
2(x2+x+1)√(x+1) 2(x2+x+1)√(x+1)
5.28.
y' = 2x√(1-x4)+2x(x2+2)/√(1-x4) = 3x-x5+x3
2-2x4 (1-x4)3/2
5.29.
y' = (√(2x-1)+(x+3)/√(2x-1))(2x+7)-(2x+6)√(2x-1) =
(2x+7)2
= (3x+2)(2x+7)-(2x+6)(2x-1) = 2x2+15x+20
(2x+7)2√(2x-1) (2x+7)2√(2x-1)
5.30.
y' = (3+1/(2√x))√(x2+2)-(3x+√x)x/√(x2+2) =
x2+2
= (6√x+1)(x2+2)-2x√x(3x+√x) = 12√x+2-x2
2√x(x2+2)3/2 2√x(x2+2)3/2
5.31.
y' = (18x5+16x3-2x)√(1+x2)-x(3x6+4x4-x2-3)/√(1+x2) = 16x7+14x5+16x4+15x3
15+15x2 15(1+x2)3/2