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Реферат: 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 )

= x 11( x 16-2 x 8+1) = ( x 8-1)2 .

x24 √(1+x8 ) x13 √(1+x8 )

5.6.

2x√(1-3x4 ) + 6 x 5 ­

y'= √(1-3 x 4 ) = 2 x (1-3 x 4 )+6 x 5 = 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