Реферат: Решения к Сборнику заданий по высшей математике Кузнецова Л.А. - 2. Дифференцирование. Зад.6

Задача 6 . Найти производную.

6.1.

e^x + 2 e ^{2 x} + e ^x

y' = 1- √( e ^{2 x} + e ^x +1) = 2+ e ^x +√( e ^{2 x} + e ^x +1)- e ^x √( e ^{2 x} + e ^x +1)-2 e ^{2 x} - e ^x =

2+e^x +2√(e^2x +e^x +1) 2+e^x +2√(e^2x +e^x +1)

= (2-e ^x )√(e ^2x +e ^x +1)+2+e ^x -2e ^x

2+e^x +2√(e^2x +e^x +1)

6.2.

y' = 1/4*e^2x (2-sin2x-cos2x)+1/8*e^2x (-2cos2x+2sin2x)=1/8*e^2x (4-2sin2x-2cos2x-2cos2x+2sin2x)=1/8*e^2x (4-4cos2x)=e^2x *sin2x

6.3.

y' = 1 * 1 * 2e ^x = e ^x .

2 1 + (e ^x -3) ² 4 e^2x -6e^x +10

4

6.4.

y' = 1 * 1-2 ^x * -2 ^x ln2(1+2 ^x )-(1-2 ^x )2 ^x ln2 = (2 ^x -1)2 ^x ln4 = 2 ^x (2 ^x -1)

ln4 1+2^x (1+2^x )² ln4(1+2^x )³ (1+2^x )³

6.5.

e ^x (√(e ^x +1)+1) _ e ^x (√(e ^x +1)-1)

y' = e ^x + √(e ^x +1)+1 * 2√(e ^x +1) 2√(e ^x +1) =

√(e^x +1) √(e^x +1)-1 (√(e^x +1)+1)²

= e ^x + e ^x √(e ^x +1)+e ^x -e ^x √(e ^x +1)+e ^x = √(e^x +1)

√(e^x +1) 2e^x √(e^x +1)

6.6.

y' = 2/3*3/2*√(arctge^x ) * e ^x = e ^x √(arctge ^x )

1+e^x 1+e^x

6.7.

y' = 2e ^x - 2e ^x = e ^x

2(e^2x +1) 1+e^2x 1+e^2x

6.8.

6.9.

y' = 2/ln2*((2^x ln2)/(2√(2^x -1))-(2^x ln2)/(1+2^x -1))=2x/√(2^x -1)-2

6.10.

e ^x (√(1+ e ^x )+1) _ e ^x (√(1+ e ^x )-1)

y'= 2√(1+e^x )+2 e ^x ( x -2) _ √(1+ e ^x )+1 * 2√(1+ e ^x ) 2√(1+ e ^x ) =

2√(1+e^x ) √(1+e^x )-1 (√(1+e^x )+1)

= xe ^x +2 _ 2e ^x √(1+e ^x )+2e ^x = xe ^x .

√(1+e^x ) e^x √(1+e^x )( √(1+e^x )+1) √(1+e^x )

6.11.

y'= αe ^αx (αsinβx-βcosβx)+e ^αx (αβcosβx+β ² sinβx) =

α² +β²

= e ^αx (α ² sinβx+β ² sinβx) = e^αx sinβx

α² +β²

6.12.

y'= αe ^αx (βsinβx-αcosβx)+e ^αx (β ² cosβx+αβsinβx) =

α² +β²

= e ^αx (β ² cosβx+2αβsinβx-α ² cosβx)

α² +β²

6.13.

y'= ae^ax * ┌ 1 + acos2bx+2bsin2bx ┐+ e^ax ┌ -2absin2bx+4b ² cos2bx ┐=

└ 2a 2(a² +4b² ) ┘ └ 2(a² +4b² ) ┘

= e^ax /2*(1+cos2bx)= e^ax cos² bx

6.14.

y' = 1 – e ^x - e ^x = 1 - e ^x -e ^x -e ^2x = 1 + e ^2x .

(1+e^x )² 1+e^x (1+e^x )² (1+e^x )²

6.15.

3/6*e^x/6 *√(1+e^x/3 ) + 1/3*e ^x/3 (1+e ^x/6 )

y'= 1 - 2√(1+e ^x/3 ) _ 3/6*e ^x/6 =

(1+e^x/6 )√(1+e^x/3 ) 1+e^x/3

= 1- e ^x/6 +e ^x/2 +e ^x/3 +e ^x/2 _ e ^x/6 = 1- e ^x/3 -e ^x/6 .

2(1+e^x/6 )(1+e^x/3 ) 2(1+e^x/3 ) 2(1+e^x/6 )(1+e^x/3 )

6.16.

y' = 1 - 8e ^x/4 = 1 - 2e ^x/4 .

4(1+e^x/4 )² (1+e^x/4 )²

6.17.

e^x + e ^2x

y'= √(e ^2x -1) ­_ e ^-x = e ^x (e ^x +√(e ^2x -1)) _ e ^-x *e ^x = e ^x -1 .

e^x +√(e^2x -1) √(1-e^-2x ) (e^x +√(e^2x -1))√(e^2x -1) √(e^2x -1) √(e^2x -1)

6.18.

e ^2x

y'= 1+e^-x arcsine^x – e ^-x *e ^x + √(1-e ^2x ) =

√(1-e^2x ) 1+√(1-e^2x )

= 1+e^-x arcsine^x - 1 + e ^2x =

√(1-e^2x ) (1+√(1-e^2x )) √(1-e^2x )

= e^-x arcsine^x

6.19.

y'= 1- e ^x +e^-x/2 arctge^x/2 – e ^-x/2 *e ^x/2 _ e^x/2 arctge^x/2 =

1+e^x 1+e^x 1+e^x

= 1- e^x + 1 + arctge^x/2 * 1-e^x = arctge^x/2 * 1-e^x .

1+e^x 1+e^x e^x/2 (1+e^x ) e^x/2 (1+e^x )

6.20.

y'= 3x² e^x3 (1+x³ )-3e^x3 x² = 3x⁵ e^x3

(1+x³ )² (1+x³ )²

6.21.

y'= b *me^mx √a = e^mx .

m√(ab)(b+ae^2mx ) √b b+ae^2mx

6.22.

y'= e^{3 ^√x} /3√x(³ √x² -2³ √x+2)+3e^{3^√x} (2/(3³ √x)-2/(3³ √x² ))= e^{3^√x}

3^√x= кубический корень из х

6.23.

( e^x +2e^2x _ e^x )(√(1+e^x +e^2x )-e^x +1) _ ( e^x +2e^2x _ e^x )(√(1+e^x +e^2x )-e^x -1)

y'= √(1+e^x +e^2x )-e^x +1 * 2√(1+e^x +e^2x ) 2√(1+e^x +e^2x ) =

√(1+e^x +e^2x )-e^x -1 (√(1+e^x +e^2x )-e^x +1)²

= e^x (1+2e^2x -2√(1+e^x +e^2x )) = 1 .

(e^x (1+2e^2x -2√(1+e^x +e^2x )))√(1+e^x +e^2x ) √(1+e^x +e^2x )

6.24.

y'= cosxe^sinx (x-1/cosx)+e^sinx (1-sinx/cos² x)= e^sinx (xcosx-sinx/cos² x)

6.25.

y'= e^x /2((x² -1)cosx+(x-1)² sinx)+e^x /2(2xcosx-(x² -1)sinx+2(x-1)sinx+(x-1)² cosx)=

= e^x /2(x-1)(5x+3)cosx

6.26.

y'= e^x +e^-x = e^3x +e^{x .}

1+(e^x -e^-x )² e^4x -e^2x +1

6.27.

y'= e^{3^√x} /³ √x² (³ √x⁵ -5³ √x⁴ +20x-60³ √x² +120³ √x-120)+e^{3^√x} (5³ √x² -20³ √x+20-120/³ √x+120/³ √x² )= e^{3^√x} (x-40)

6.28.

y'= -3e^3x sh³ x+3e^3x sh² xchx = e^3x (chx-shx)

3sh⁶ x sh⁴ x

6.29.

y'= -e^-x + e^2x = √(e^4x -e^2x )-√(e^-2x -1) = √(e^2x -1)-√(1-e^2x )

√(1-e^-2x ) √(1-e^2x ) √(1-e^-2x )*√(1-e^2x ) √(1-e^-2x ) √(1-e^2x )

6.30.

y'= xe^-x2 (x⁴ +2x² +2)-1/2*e^-x2 (4x³ +4x)= x⁵ e^-x2

6.31.

y'= 2xe^x2 (1+x² )-2e^x2 x = 2x³ e^x2

(1+x² )² (1+x² )²