Derivadas Resueltas

PROBLEMAS
Hallar dydx para cada una de las funciones siguientes:
1.- y = u6, u= 1 + 2 x
= ( 1+2x)6
y’ = 6 ( 1+2x)5 ddx ( 1+2x)
y’ = 6 ( 1+2x)5 [ddx (1) + 2 ddx (x)12]
y’ = 6x ( 1+x)5
dydx= 6 u5x

2.- y= 2 u- u2, u= x3 - x
= 2 (x3-x) – (x3- x)2
y'= ddx [2 x3-x]12- ddx (x3- x)2
y’= 12 [2 x3-x]-12 - ddx [2 x3-x]- [2 x3-x ddx (x3- x)]y’= 12 2 x3-x [ 2 (3x2 - 1) ] – {[2 x3-x] (3x2 - 1)}
y’= [12 x3-x - 2 x3-x] (3x2 - 1)
dydx= 12 u – 2 u (3x2 - 1)

3.- y= a-ua+u , u= b-xb+x
y’= a+u ddx a-u –[a-u ddx a+u](a+u)2
y’= a+u ddx-u- [ a-uddx u](a+(a+u)2u)2
y’= - a+uddx u- [ a-ud dx u](a+u)2
y’= ddx (u)(a+u)2- [ -a+u- a-u]
y’= ddx ua+u2 -a-u-a+u =ddx ua+u2 -2a
y’= -2 bb+x2a+u21 -2 a =-2b(a+u)2 (b+x)2- 2 a
dydx = 4b(a+u)2 (b+x)2

4.- y= u a2- u2 , u= 1- x2
y’= uddx a2- u2+ a2- u2 ddxu
y’= u [ 12 a2- u2 ddx a2- u2]+ a2- u2 (ddxu )
y’= u 12 a2- u2 -2uddxu+ a2- u2 (ddxu )
y’= (-2u22a2- u2) ddxu + a2- u2 (ddxu )
y’= ddxu [- u2a2- u2 + a2- u2 ]
y’= ddxu [ a2- u21 - u2a2- u2 ]
y’= ddxu [ a2-u2-u2a2- u2 ]
y’= ddxu [ a2-2u2a2- u2]
u= 1- x2
∴ y'= ddx (1- x2 ) [ a2-2u2a2- u2 ]
y’= (ddx 1-x2)12 [ a2-2u2a2- u2 ]
y’= (121- x2 ) (-2x) [ a2-2u2a2- u2 ]

y’= - x1- x2 [ a2-2u2a2- u2 ]
y’= x (2u2- a2)(1- x2)(a2- u2)
dydx= x (2u2- a2)(a2- u2)(1- x2)

5.- 15x= 15y + 5y3+ 3 y5
x= 15y+ 5y3+ 3y515
x= y +13 y3+ 15 y5
x’= 1y'= 1ddx y= dydx
dydx=1+ y2+ y4
∴dydx = 1dydx= 11+ y2+ y4

6.- x= y+ 3y
x= y12 + y13
x'=1y'= 1dydx= dxdy
dxdy= 12 y-12+ 13 y-23
dydx = 1dxdy= 112y12+13y23
dydx= 13y23 + 2y122y12 3y23=6 y12y233y23 +2y12
dydx= (y12) (6y23)(y12) (3 y16+ 2 )= 6y233 y16+ 2

7.- y2=2 px
y= 2 px
y'=122 px (2p)
y'=p2 px

8.- x2 + y2 = r2
y2= r2- x2
y=r2- x2
y'=ddx(r2- x2 )
y'=(12 r2- x2 ) (- 2 x)
y'= -xr2- x2

9.- b2x2+ a2y2= a2 b2
a2y2= a2 b2- b2x2
y2= a2 b2-b2x2 a2
y=a2 b2-b2x2 a2
y= (a2 b2- b2x2)12a
y'=a2 a2 b2- b2x2)-12 ddx (a2 b2- b2x2- (a2 b2- b2x2)12 a2
y'= a2a21 (a2 b2- b2x2)-12 (-2b2x)
y'= a2a2 (a2 b2- b2x2)-12 (-2b2x)
y'= - 2 b2xaa2 b2-b2x2


10.- x+ y = a
y= (a - x )2y'=2 (a - x ) ddx (a - x )
y'= 2 (a - x ) (- 12 x )
y'= x- ax

11.- x23+ y23 = a23
y23= a23 - x23
y=(a23 - x23)23
y'=23 (a23 - x23)-13 (-23 x-13)
y'= - 4x-139 (a23 - x23)13
y'= - 493 a23 - x23* 3x
y'= - 49 3x( a23 - x23)

12.- x3- 3axy+ y3= 0
- 3axy + y3= -x3
y3= -x3 + 3axy
Derivando implícitamente:3y2 * y’= -3x2 + 3 a (xy'+y)
3y2 * y’= -3x2+ 3axy'+ 3 ay
3y2*y'- 3axy'= -3x2+ 3 ay
y' 3y2- 3ax= -3x2+ 3 ay
y'= -3x2+ 3 ay3y2- 3 ax
Factorizando el signo menos
y'= -(3x2- 3 ay)-(3ax- 3 y2)
y'= 3x2- 3 ay3ax- 3 y2

13.- x3 + 3x2y+ y3= c3
y3= c3- 3x2y –x3
Derivando implícitamente
3y2y'= -3 x2y'+ y 2x- 3x2
3y2y'= -3x2y'- 6 xy - 3x2
3y2y'+ 3x2y'= -6xy - 3x2
y' 3y2+ 3x2= -6xy -3x2
y'= -6xy - 3x23y2+ 3x2
y'= - 3x2+ 6 xy3x2+ 3y2

14.- x+2xy+ y=a
y=a-x- 2xy

Derivando implícitamente
y'=-1-2 [(12 xy) (xy’ + y )]
y'=-1-2 [ xy'+ y2 xy]
y'=-1-xy'+ yxy
y'= -xy- xy'+yxy
y'=xy+ xy'= -xy- y
y'=xy+ x= -xy- y
y'= - y+ xyx+ xy

15.- x2+ a xy+ y2= b2
y2= b2- x2- a xy
Derivando implícitamente
2y y'= -2x-a 12xyxy'+ y
2y y'= -2x-a xy'+ y2 xy
2y y'= -2x+ -axy'-ay2 xy
2y y'= -2x- ay+axy'2xy
2y y'= -2x- axy'2xy- ay2xy
2y y'= -2x –axy'2xy –ay2xy
2y y'+ axy'2xy= -2x-ay2xy
y'2y+ ax2xy= -2x-ay2xy
y'= -2x1- ay2xy2y1+ ax2 xy
y'=-2x 2xy- ay2 xy2y 2xy+ ax2 xy
y'= - 2x 2xy+ ay 2y 2xy+ ax
y'= - 4x xy+ ay4y xy+ ax

16.- x4+ 4x3y+y4=20
y4=20-x4- 4x3y
4y3y'= -4x3- 4 x3y'+ 3x2y
4y3y'= -4x3-4x3y'- 12 x2y
4y3y'+ 4x3 y'= -4x3- 12x2y
y'4y3+4x3= -4x3-...