Solucion_guia_8[1]
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SOLUCION GUIA 8
1. x5 = 5x4
= x ( 3 ) = ( 3) 2 x 1
3
2. x 3. 1
=
t3
1
2
( ) = − 3t
t 3 ( 0 ) − 1 3t 2
(t )
3 2
2
t6
( )
4.
4 u 4 ( 0 ) − 4 4u 3 − 16u 3 = = 2 u4 u8 u4
5.
1 5u 5 ( 0) − 1 25u 4 − 25u 4 = = 2 5u 5 5u10 5u 5
( )
( ( )
)
( )
7 6 7 6 6. x = 7 7 x − x ( 0 ) = 49 x 2
( 7)
7
1
7. 3
x
2
=
1 x
2
3
=
49
x
2
3
( 0) − 1 2 3 x
x 3 2
−1
2
3
− 2 −13 x = 3 4 x 3
8.
2x - x³ = -x³ + 2x = -3x² + 2
9.
4x³ - 3x² + 7= 12x² - 6x
10. 5 – 2x² + x⁴ = x⁴ - 2x² + 5 = 4x³ - 4x 11. 3x⁴ - 7x³ + 5x² + 8 = 12x³ - 21x² + 10x 12. 4 x 3 + 2 + 1 = 12 x 2 + x( 0 ) − 1(1) = 12 x 2 + 1 2 2
x
( x)
x
2 13. 3u 2 + 3 = 6u + u ( 0) − 3( 2u ) = 6u + − 6 2 2 4 u
(u ) 2
( )
u
( )
6 5 6 6 5 5 5 14. x + 6 = 6 6 x − x ( 0 ) + x ( 0 ) − 6 6 x = 36 x + 36 x 2 2 6 12
6
x
(x )
( 6)
36
6
(
x
)
− 0.4 − 0.4 0.6 15. x1.2 + 1 = 1.2 x 0.2 + x ( 0) − 1 0.6 x = 12 x 0.2 + − 1.6 x x1.2 0.6 2 x 0.6
(x )
(
16. x 0.4 − x −0.4 = 0.4 x −0.6 − − 0.4 x −1.4
−3 1 1 −1 −1 2 2 = 2 x 2 + 1 = 2 x 4 + 2 x 2 = x 2 + x 2 x x 2
17. 2 x +
18. x 7 +
(
2 x3
= 2x
3
2
2 + 3 x 2
3 1 3 = 2t 2 − 1 t t 3
20. 2 t − 3
22.
3
2
+ 4x
x−3
3
)
(
1 7 + 7 x + + 7 = 7 x 6 + 1x −7 + 7 = 7 x 6 + − 7 x −8 7 x x
3 19. 2 x +
21. 2 x
)
5
4
= 4x
5
4
)
3 −3 −3 1 = 2 x 2 + 2 x 2 = 3x 2 + − 3x 2
−3 −1 1 −1 = 2t 2 − 3t 3 = 2 x 2 − − 3t 3
+ 2x
3
2
= 20
4
4
+6
2
2
1 1 = x 13 − x −13 = 1 x −2 3 − − x −4 3 = 1 x− 3 1 x 3 x 3
23. 3x4 + (2x – 1)² = 12x³ + 2(2x – 1) (2) = 12x³ + 4(2x – 1) 24. (y – 2) (2y – 3) = (y – 2) (2) – (2y – 3) (1) = (2y – 4) – (2y – 3) 25. (x – 7) (2x – 9) = (x – 7) (2) – (2x – 9) (1) = (2x – 14) – (2x – 9) 26. x + 1 = 2 x + 1 1 + x( 0 ) − 1(1) 2 2
x
x
( x)
1 − 1 = 2 x + 1 + 2 x ( x )
27. (u + 1) (2u + 1) = (u + 1) (2) – (2u + 1) (1) = (2u + 2) – (2u + 1) 2
1 −1 1 1 1 x+ = 2 x 2 + 1 x 2 + x 2 x x 2 28. −3 −3 1 1 = 2 x 2 + 1 1 x 2 + − 1 x 2 2 2 x 2
1 1 = 2 x 2 + 1 x 2
29. (t + 1) (3t – 1)² = (t + 1) 2(3t – 1) (3) – (3t – 1)² (1) = (t + 1) 6 (3t – 1) – (3t – 1)² = (t + 1) (18t – 6) – (3t – 1)² 30. (u – 2)³ = 3 (u – 2)² (1) = 3(u – 2)² 31. (x + 2)³ = 3(x + 2)² (1) = 3(x + 2)² 32. (x + 1) (x – 1)² = (x + 1) 2 (x – 1) (1) – (x + 1)² (1) = (x + 1) 2 (x – 1) – (x + 1)² = (x + 1) (2x – 2) – (x + 1)² 33.
3
x +1 x +1 = 3 x x
2
x(1) − ( x + 1)(1) 2 ( x )
x + 1 x − ( x + 1) = 3 x2 x 2
34. 2t − 1 3
2t − 1 = 3 2t 2t
2
2t ( 2 ) − 2t − 1( 2 ) ( 2t ) 2
2t − 1 4t − ( 4t − 2 ) = 3 4t 2 2t 2
3
y+2 y−2 + y y 35.
3
y+2 = 3 y
2
y (1) − ( y + 2)(1) y − 2 + 3 y2 y
y+2 = 3 y
2
y − ( y + 2) y − 2 + 3 y2 y
2
2
y (1) − ( y − 2)(1) y
y − ( y − 2) y
36. 2 y 2 + 3 y − 7
y =
y ( 4 y )( 3) − ( 2 y + 3 y − 7 )(1) y2
4 y 2 ( 3) − ( 2 y + 3 y − 7 ) = y2
=
12 y 2 − ( 2 y + 3 y − 7 ) y2
( x + 1)
37.
x x( 2( x + 1) ) − ( x + 1) (1) = x2 2
x( 2( x + 1) ) − ( x + 1) = x2 =
38.
2
2 x( x + 1) − ( c + 1) x2
x 2 − 3x + 1 x
=
=
x
1
2
( 2 x − 3) − ( x 2 − 3x + 1) 1 2 x 2 (1) −1
x 2 1
2x
1
2
− 3x
1
2
(
2
)
−1 − x 2 − 3 x + 1 1 x 2 2 x
39.
t+3
t
t
=
t+ t
1
3 t 2
3 −1 = t + t 2 t −3 2 1 + t ( 0 ) − 3(1) − 1 t 2 = 2 t
−3 −3 = 2 − 1 t 2 t 2
( x + 1) 2 + ( x − 1) 2 x2
40.
=
=
=
(
x 2 ( 2( x + 1)(1) ) + ( 2( x + 1)(1) ) − ( x + 1) + ( x − 1) ( 2 x )
(x )
2 2
2
(
2
) (
2
2
x 2 ( 2( x + 1) ) + ( 2( x + 1) ) − ( 2 x + 2 ) + ( 2 x − 1) x4
((2x
3
)
+ 2 x 2 + ( 2 x + 1) − ( 2 x + 2 ) + ( x − 1) x4
2
)
2
)
)
41.
( 2t + 3) 2 − ( 2t − 3) 2 4t 4( 2( 2t + 3)( 2 ) ) − ( 2( 2t − 3)( 2) ) − ( 2t + 3) − 2( − 3) ( 4t ) 2
=
4( 4( 2t + 3) ) − ( 4( 2t − 3) ) − ( 8t + 3) − ( 2t − 3) 4t 2
=
4( ( 8t + 12 ) − ( 8t − 12 ) ) − ( 8t + 3) − ( 2t − 3) 4t 2
2
(
(
2
2
x1.6 x − 2.3 x 3
(
) ( )( ( )
x 2.3 1.6 x 0.6 − x1.6 2.3x1.5 = 3 x − x 2.3 2
(
) (
1.61.38 − 2.3 x 2.08 = 3 x 2 − x 4.6 43.
2
( 32t + 48) − ( 8t − 12) − (( 8t + 3) 2 − ( 2t − 3) 2 )
= 42.
(
=
2 y + (3y) = 2y =1 =y
1
2 −1
2
−1
(
+ − 1( 3 y )
−2
( 3) )
( 2 y ) −12 + ( − 3( 3 y ) −2 ) 2
+ ( − 9 y)
−2
)
)
)
2
)
2
)( 4 )
44.
(8 y) =2
45.
2
3
+ (8 y)
−2
3
−1 ( 8 y ) 3 ( 8) + − 2 ( 8 y ) 3 ( 8) 3 3 −5
= 116
(8 y) 3
(16t )
4
3
−1
3
− (16t )
+ − 16
−3
(8 y) 3
−5
3
4
) (
(
−7 −1 = 3 (16t ) 4 (16) − − 3 (16t ) 4 (16) 4 4
= 48 (16t ) 4 46.
3
47.
2
3
4
(
−7 − − 48 (16t ) 4 4
3
27t 2
1 − 2 27t 3
= 2 27t 3
−1
= 2 27t 3
−1
=dy2 27t 3 dy
−1
y = x3 +
)
1
27t 2 −
= 27t
−1
3
3
3
−2 − 1 27t 3
− − 2 27t 3
−5
3
( 27 )
−5 − − 18 27t 3
1 x3
( )
x3 ( 0) − 1 3x 2 = y = 3x 2 − 2 x3
( )
− 3x 2 = y = 3 x + 6 x 2
)
48.
du dy Siu = x 2 − 7 x +
5 x
x( 0 ) − 5(1) = u = 2x − 7 + x2 −5 = u = 2x − 7 + 2 x 49.
dy = du Siy = u 3 − 5u 2 +
7 +6 3u 2
3u 2 ( 0 ) − 7( 6u ) = y = 3u 2 − 10u + 2 2 3 u
( )
− 42u = y = 3u 2 − 10u + 4 3u
(
)
dy t 3 − 5t 2 + 7t − 1 = du t2 50.
( (
) ( ( )
(
) (
))
=
dy t 2 3t 2 − 10t + 7 − t 3 − 5t 2 + 7 − 1( 2t ) = 2 du t2
=
dy 3t 4 − 10t 3 +7t 2 − 2t 4 − 10t 3 + 14t 2 − 2t = du t4
=
dy 3t 4 − 10t 3 + 7t 2 − 2t 4 + 10t 3 − 14t 2 + 2t = du t4
=
dy t 4 − 7t 2 + 2t = du t4
)
dy x , pruebe que 2 y = 1 dx
51. Si Y=
=2 x 1 −1 = 2 x 2 1 x 2 2
= 1x 0 = 1
52. Si
1 dy , pruebe que 2 y +1 = 0 x dx
u=
1 = 2 x
1 = 2 1 x 2
−3
12 1 −12 x ( 0 ) − 1 2 x (1) +1 = 0 2 1 2 x
− 1 x −1 2 −1 = 2 x 2 2 +1 = 0 x −1 = 2 x 2
1. x5 = 5x4
= x ( 3 ) = ( 3) 2 x 1
3
2. x 3. 1
=
t3
1
2
( ) = − 3t
t 3 ( 0 ) − 1 3t 2
(t )
3 2
2
t6
( )
4.
4 u 4 ( 0 ) − 4 4u 3 − 16u 3 = = 2 u4 u8 u4
5.
1 5u 5 ( 0) − 1 25u 4 − 25u 4 = = 2 5u 5 5u10 5u 5
( )
( ( )
)
( )
7 6 7 6 6. x = 7 7 x − x ( 0 ) = 49 x 2
( 7)
7
1
7. 3
x
2
=
1 x
2
3
=
49
x
2
3
( 0) − 1 2 3 x
x 3 2
−1
2
3
− 2 −13 x = 3 4 x 3
8.
2x - x³ = -x³ + 2x = -3x² + 2
9.
4x³ - 3x² + 7= 12x² - 6x
10. 5 – 2x² + x⁴ = x⁴ - 2x² + 5 = 4x³ - 4x 11. 3x⁴ - 7x³ + 5x² + 8 = 12x³ - 21x² + 10x 12. 4 x 3 + 2 + 1 = 12 x 2 + x( 0 ) − 1(1) = 12 x 2 + 1 2 2
x
( x)
x
2 13. 3u 2 + 3 = 6u + u ( 0) − 3( 2u ) = 6u + − 6 2 2 4 u
(u ) 2
( )
u
( )
6 5 6 6 5 5 5 14. x + 6 = 6 6 x − x ( 0 ) + x ( 0 ) − 6 6 x = 36 x + 36 x 2 2 6 12
6
x
(x )
( 6)
36
6
(
x
)
− 0.4 − 0.4 0.6 15. x1.2 + 1 = 1.2 x 0.2 + x ( 0) − 1 0.6 x = 12 x 0.2 + − 1.6 x x1.2 0.6 2 x 0.6
(x )
(
16. x 0.4 − x −0.4 = 0.4 x −0.6 − − 0.4 x −1.4
−3 1 1 −1 −1 2 2 = 2 x 2 + 1 = 2 x 4 + 2 x 2 = x 2 + x 2 x x 2
17. 2 x +
18. x 7 +
(
2 x3
= 2x
3
2
2 + 3 x 2
3 1 3 = 2t 2 − 1 t t 3
20. 2 t − 3
22.
3
2
+ 4x
x−3
3
)
(
1 7 + 7 x + + 7 = 7 x 6 + 1x −7 + 7 = 7 x 6 + − 7 x −8 7 x x
3 19. 2 x +
21. 2 x
)
5
4
= 4x
5
4
)
3 −3 −3 1 = 2 x 2 + 2 x 2 = 3x 2 + − 3x 2
−3 −1 1 −1 = 2t 2 − 3t 3 = 2 x 2 − − 3t 3
+ 2x
3
2
= 20
4
4
+6
2
2
1 1 = x 13 − x −13 = 1 x −2 3 − − x −4 3 = 1 x− 3 1 x 3 x 3
23. 3x4 + (2x – 1)² = 12x³ + 2(2x – 1) (2) = 12x³ + 4(2x – 1) 24. (y – 2) (2y – 3) = (y – 2) (2) – (2y – 3) (1) = (2y – 4) – (2y – 3) 25. (x – 7) (2x – 9) = (x – 7) (2) – (2x – 9) (1) = (2x – 14) – (2x – 9) 26. x + 1 = 2 x + 1 1 + x( 0 ) − 1(1) 2 2
x
x
( x)
1 − 1 = 2 x + 1 + 2 x ( x )
27. (u + 1) (2u + 1) = (u + 1) (2) – (2u + 1) (1) = (2u + 2) – (2u + 1) 2
1 −1 1 1 1 x+ = 2 x 2 + 1 x 2 + x 2 x x 2 28. −3 −3 1 1 = 2 x 2 + 1 1 x 2 + − 1 x 2 2 2 x 2
1 1 = 2 x 2 + 1 x 2
29. (t + 1) (3t – 1)² = (t + 1) 2(3t – 1) (3) – (3t – 1)² (1) = (t + 1) 6 (3t – 1) – (3t – 1)² = (t + 1) (18t – 6) – (3t – 1)² 30. (u – 2)³ = 3 (u – 2)² (1) = 3(u – 2)² 31. (x + 2)³ = 3(x + 2)² (1) = 3(x + 2)² 32. (x + 1) (x – 1)² = (x + 1) 2 (x – 1) (1) – (x + 1)² (1) = (x + 1) 2 (x – 1) – (x + 1)² = (x + 1) (2x – 2) – (x + 1)² 33.
3
x +1 x +1 = 3 x x
2
x(1) − ( x + 1)(1) 2 ( x )
x + 1 x − ( x + 1) = 3 x2 x 2
34. 2t − 1 3
2t − 1 = 3 2t 2t
2
2t ( 2 ) − 2t − 1( 2 ) ( 2t ) 2
2t − 1 4t − ( 4t − 2 ) = 3 4t 2 2t 2
3
y+2 y−2 + y y 35.
3
y+2 = 3 y
2
y (1) − ( y + 2)(1) y − 2 + 3 y2 y
y+2 = 3 y
2
y − ( y + 2) y − 2 + 3 y2 y
2
2
y (1) − ( y − 2)(1) y
y − ( y − 2) y
36. 2 y 2 + 3 y − 7
y =
y ( 4 y )( 3) − ( 2 y + 3 y − 7 )(1) y2
4 y 2 ( 3) − ( 2 y + 3 y − 7 ) = y2
=
12 y 2 − ( 2 y + 3 y − 7 ) y2
( x + 1)
37.
x x( 2( x + 1) ) − ( x + 1) (1) = x2 2
x( 2( x + 1) ) − ( x + 1) = x2 =
38.
2
2 x( x + 1) − ( c + 1) x2
x 2 − 3x + 1 x
=
=
x
1
2
( 2 x − 3) − ( x 2 − 3x + 1) 1 2 x 2 (1) −1
x 2 1
2x
1
2
− 3x
1
2
(
2
)
−1 − x 2 − 3 x + 1 1 x 2 2 x
39.
t+3
t
t
=
t+ t
1
3 t 2
3 −1 = t + t 2 t −3 2 1 + t ( 0 ) − 3(1) − 1 t 2 = 2 t
−3 −3 = 2 − 1 t 2 t 2
( x + 1) 2 + ( x − 1) 2 x2
40.
=
=
=
(
x 2 ( 2( x + 1)(1) ) + ( 2( x + 1)(1) ) − ( x + 1) + ( x − 1) ( 2 x )
(x )
2 2
2
(
2
) (
2
2
x 2 ( 2( x + 1) ) + ( 2( x + 1) ) − ( 2 x + 2 ) + ( 2 x − 1) x4
((2x
3
)
+ 2 x 2 + ( 2 x + 1) − ( 2 x + 2 ) + ( x − 1) x4
2
)
2
)
)
41.
( 2t + 3) 2 − ( 2t − 3) 2 4t 4( 2( 2t + 3)( 2 ) ) − ( 2( 2t − 3)( 2) ) − ( 2t + 3) − 2( − 3) ( 4t ) 2
=
4( 4( 2t + 3) ) − ( 4( 2t − 3) ) − ( 8t + 3) − ( 2t − 3) 4t 2
=
4( ( 8t + 12 ) − ( 8t − 12 ) ) − ( 8t + 3) − ( 2t − 3) 4t 2
2
(
(
2
2
x1.6 x − 2.3 x 3
(
) ( )( ( )
x 2.3 1.6 x 0.6 − x1.6 2.3x1.5 = 3 x − x 2.3 2
(
) (
1.61.38 − 2.3 x 2.08 = 3 x 2 − x 4.6 43.
2
( 32t + 48) − ( 8t − 12) − (( 8t + 3) 2 − ( 2t − 3) 2 )
= 42.
(
=
2 y + (3y) = 2y =1 =y
1
2 −1
2
−1
(
+ − 1( 3 y )
−2
( 3) )
( 2 y ) −12 + ( − 3( 3 y ) −2 ) 2
+ ( − 9 y)
−2
)
)
)
2
)
2
)( 4 )
44.
(8 y) =2
45.
2
3
+ (8 y)
−2
3
−1 ( 8 y ) 3 ( 8) + − 2 ( 8 y ) 3 ( 8) 3 3 −5
= 116
(8 y) 3
(16t )
4
3
−1
3
− (16t )
+ − 16
−3
(8 y) 3
−5
3
4
) (
(
−7 −1 = 3 (16t ) 4 (16) − − 3 (16t ) 4 (16) 4 4
= 48 (16t ) 4 46.
3
47.
2
3
4
(
−7 − − 48 (16t ) 4 4
3
27t 2
1 − 2 27t 3
= 2 27t 3
−1
= 2 27t 3
−1
=dy2 27t 3 dy
−1
y = x3 +
)
1
27t 2 −
= 27t
−1
3
3
3
−2 − 1 27t 3
− − 2 27t 3
−5
3
( 27 )
−5 − − 18 27t 3
1 x3
( )
x3 ( 0) − 1 3x 2 = y = 3x 2 − 2 x3
( )
− 3x 2 = y = 3 x + 6 x 2
)
48.
du dy Siu = x 2 − 7 x +
5 x
x( 0 ) − 5(1) = u = 2x − 7 + x2 −5 = u = 2x − 7 + 2 x 49.
dy = du Siy = u 3 − 5u 2 +
7 +6 3u 2
3u 2 ( 0 ) − 7( 6u ) = y = 3u 2 − 10u + 2 2 3 u
( )
− 42u = y = 3u 2 − 10u + 4 3u
(
)
dy t 3 − 5t 2 + 7t − 1 = du t2 50.
( (
) ( ( )
(
) (
))
=
dy t 2 3t 2 − 10t + 7 − t 3 − 5t 2 + 7 − 1( 2t ) = 2 du t2
=
dy 3t 4 − 10t 3 +7t 2 − 2t 4 − 10t 3 + 14t 2 − 2t = du t4
=
dy 3t 4 − 10t 3 + 7t 2 − 2t 4 + 10t 3 − 14t 2 + 2t = du t4
=
dy t 4 − 7t 2 + 2t = du t4
)
dy x , pruebe que 2 y = 1 dx
51. Si Y=
=2 x 1 −1 = 2 x 2 1 x 2 2
= 1x 0 = 1
52. Si
1 dy , pruebe que 2 y +1 = 0 x dx
u=
1 = 2 x
1 = 2 1 x 2
−3
12 1 −12 x ( 0 ) − 1 2 x (1) +1 = 0 2 1 2 x
− 1 x −1 2 −1 = 2 x 2 2 +1 = 0 x −1 = 2 x 2
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