guia 1 pep2 derivacion recta tangente y normal

7/13/2019 Guia 1 Pep2 Derivacion Recta Tangente y Normal

1/10

y= f(x)

f(x) = ln x sin x+ tan xx2

f(x) = sin x

x + ln x cosx+ x

2 sec2 x 2x tanxx4

= sin x

x + ln x cosx+ x sec

2 x 2tanxx3

f(x) f(x) = ln

etanx2x3+5

f(x) = ln etanx ln

2x3 + 5

= tan x 12

ln(2x3 + 5)

f(x) = sec2 x 6x22(2x3 + 5)

= sec2 x 3x2

(2x3 + 5)

f(x) f(x) = 3

3x+ 1sinxcos(lnx)


2/10

f(x) = 2

3 3

3x+ 1sinxcos(lnx)

2

d

dx3

x+

1 sinx

cos(lnx)

= 2

3 3

3x+ 1sinxcos(lnx)

2 3

2x

+ cosx cos(lnx) + (1 sin x)

sin(lnx)

x

cos2(lnx)

= 2

3 3

3x+ 1sinx

cos(lnx)

2

2

3x

+(1 sinx) sin(lnx) x cos x cos(lnx)

x cos2(lnx)

M y= Mx2e2x d2y

dx24dy

dx+4y= 6e2x

y y

y = Mx2e2x

dy

dx = M(2xe2x + 2x2e2x)

d2y

dx2 = M

2(e2x + 2xe2x) + 2(2xe2x + 2x2e2x)

= 2Me2x

1 + 4x+ 2x2

d2y

dx2 4dy

dx+ 4y = 2Me2x(1 + 4x+ 2x2) 4Me2x(2x+ 2x2) + 4Mx2e2x

= Me2x(2 + 8x+ 4x2 8x 8x2 + 4x2)= 2Me2x


3/10

d2y

dx2 4dy

dx+ 4y= 6e2x

2Me2x = 6e2x

M = 3

f(x) = ln x cosx+ tanx arcsinx f(x) = secx1+expx+ cos x log2x f(x) = ln (1 + sinx) arctan(1 + x)

f(x) =

ln (sin(1 + (2x3 + 1)))

f(x) =

1+ln(x)

3+exp(3(x1)2) ln(sin x)

x(t) =x0cos

k

mt

k m

d2x

dt2 +

k

mx = 0

dy

dx

y+xy3

= 2 + xy


4/10

y= y(x)

y+y3 + 3xy2y = 1

2xy (1 +xy)

y+ 3xy2y xy

2xy

= 1

2xy y3

y =

1

2xyy3

1 + 3xy2 +x

2y

sin(x2 +y) 2xy3 = arctan(y x) + 2 dydx

cos(x2 +y) (2x+y) 2(y3 + 3xy2y) = 11 + (y x)2 (y

1)

2x cos(x2 +y) +y cos(x2 +y) 2y3 6xy2y = y

1 + (y x)21

1 + (y x)2

y

cos(x2 +y) 6xy2 11 + (y x)2

= 2y3 1

1 + (y x)2 2x cos(x2 +y)

y =2y3 11+(yx)2 2x cos(x2 +y)cos(x2 +y)

6xy2

1

1+(yx)2

dy

dx

(xy+ 2)2 = 2x2y2

4sin(y) cos(2x) = 1 + sin(xy)

ln(1 + 3ey) + tan(xy) = sin(3x2)

dy

dx y= xx

2


5/10

f(x) = (u(x))v(x)

ln()

ln y= x2 lnx

1

yy= 2x lnx+x

y= xx2

(2x lnx+x)

dy

dx y= (ln(1 + tanx))(1sinx)

f(x) = (u(x))v(x)

ln y= (1 sinx) ln(ln(1 + tanx))1

yy= cosx ln(ln(1 + tanx)) + (1 sin x)sec

2 x

(1 + tan x) ln(1 + tanx)

y= (ln(1 + tan

x))

(1sinx)

cos x ln(ln(1 + tan x)) + (1

sinx)sec2 x

(1 + tan x) ln(1 + tanx)

dy

dx y= (2x+ 4)sinx

dy

dx y= (exp (2(1 x2)))arctan x exp(x) =ex

f(x) =x2 +x

x0 Dom(f)


6/10

y y0= f(x0) (x x0)

y0 = f(x0)

y y0= 1

f(x0) (x x0)

y0 = f(x0)

= x20+x0

f(x) = 2x+ 1

f(x0) = 2x0+ 1

LT : y (x20+x0) = (2x0+ 1)(x x0)y = (2x0+ 1)(x x0) +x20+x0

= 2x0x 2x20+x x0+x20+x0y = x(2x0+ 1) x20


7/10

LN : y (x20+x0) = 1

(2x0+ 1) (x x0)

y = x2x0+ 1

+ x0

2x0+ 1+ x20+x0

=

x

2x0+ 1+x0+ (2x0+ 1)(x

20+x0)

2x0+ 1

= x2x0+ 1

+x0+ 2x

30+ 2x

20+x

20+x0

2x0+ 1

y = x2x0+ 1

+2x30+ 3x

20+ 2x0

2x0+ 1

x0

A(2,3)

3 = 2(2x0+ 1) x203 = 2x0+ 2 x20

x20

2x0

5 = 0

x0 = 2 4 + 20

2

x0 = 2 2

6

2

x0 = 1

6

f(x) =x3 3 (0,

2)


8/10

y = mx+n m

n

y = mx+n

y = 3x+n

(0,2)

2 = 3 0 +nn = 2

yT = 3x 2

(x0, y0) dfdx(x0, y0) = 3

f(x0) = 3x20

3 = 3x20

x01 = 1

x02 =

1

x0

f(1) = 1

yT(1) = 1

f(1) = 1yT(1) = 3 (1) 1 = 4

(1, 1)

b f(x) =


9/10

b2x3 +bx2 + 3x+ 9 x= 1 x= 2

f(x) = 3b2x2 + 2bx+ 3

f(1) = 3b2 + 2b+ 3

f(2) = 12b2 + 4b+ 3

3b2 + 2b+ 3 = 12b2 + 4b+ 3

9b2 + 2b = 0

b(9b+ 2) = 0

b1 = 0

b2 =

29

x2

a2+ y

2

b2 = 1

P(x0, y0) xx0

a2 + yy0

b2 = 1

2x

a2+ 2

yy

b2 = 0

yy

b2 = x

a2

y = b2x

a2y


10/10

(x0, y0)

mT = b2x0

a2y0

y y0 = b2

x0a2y0

(x x0)

y y0 = b2xx0

a2y0+b2x20a2y0

a2yy0 a2y20 = b2xx0+b2x20a2yy0+b

2xx0 = a2y20+ b

2x20xx0

a2 +

yy0

b2 =

x0a

2+y0b2

2xx0

a2 +

yy0

b2 = 1

f(x) = arcsin lnx1+ex

(1, f(1))

x2 +xy+y2 = 1 P0 = (2, 3)

y2 +x2y2 + cos(y2) =x 2 + sin x 2, 0

f(x) = 2x3 3x2 12x+ 20

X

y= 1 x24

y= 12x

c f(x) = c

x+ 1 (0, 3)

(5,2)

x2 +xy + y2 = 7 X

guia 1 pep2 derivacion recta tangente y normal

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