Luis Requena CI: 23780868 PROBLEMAR1O TERCER CORTE DER1VADA5

Ejercicios Propuestos Sobre Derivación

1. Use la definición de la derivada para calcular la derivada de las siguientes funciones:

a.R= f(x)=√ x f(x+h)=√ x+h f(x)=lim f ( x+h )−f (x )

h

f(x)=lim √x+h−√ xh

. √x+h+√x√x+h+√x

f(x)=lim (√ x+h )2−¿¿

f(x)=lim x+h−xh¿¿

f(x)=lim 1√x+h+√x

= 1√x+√ x

f(x)= 12√ x

b.

R=g(t+h)=(T+h) 2=t2+2th+h2

f ( x ) = l i m t2+2 th+h2−t 2

h

= l i m h(2 t+h)h

= l i m 2 t + h =

f ( x ) = 2 t

c.

Luis Requena CI: 23780868R= h(x)=

1x

h(x+h)=1x+h

f(x) =lim1x+h

−1x

h

f(x)=lim

−h( x+h ) xh1

f(x)=lim−h

hx (x+h)

f(x)=lim−1x(x+h)

f(x)= -1x . x=-

1x2

d. t(x)=

R= t(x+h=[x2+2xh+h 2+x+h+1] 3

f ( x ) = l i mx2+2xh+h2+ x+h+1¿3−¿¿

f ( x ) = l i m

x2+2 xh+h2+x+h+1−x2−x−1h

f ( x ) = l i m h(2 x+h)h

f ( x ) = 3 (x2+x+1) 2(2x+1)

f(x)=(6x+3)(x 2+x+1) 2

Luis Requena CI: 237808682. Usando las reglas de derivación, calcular la derivada de las siguientes funciones: 

a.R= h(x)=2x+3

b.

R= g(x)=9x5-12x 2+2c . g(t)=(t 2+t)(3-2t)

R=g(t)=(2t+1)(3-2t)+(t 2+t)(-2) g(t)=6t-4t2+3-2t-2t2-2tg(t)=-6t2+2t+3

d.t(x)= 5x+2 x2

x3+1

t(x)= (5∓4 x )(x3+1)−¿¿

t(x)=5x3+5+4 x4+4 x−15 x3−6 x4¿¿

t(x)=5x3+5+4 x4+4 x−15 x3−6 x4¿¿

t(x)=−2x4−10 x3+4 x+5¿¿

e. m(t)= 2tt3+2 t−4

m(t)=2 (t 3+2 t−4 )−2 t (3 t2+2)¿¿

m(t)=2t 3+4 t−8−6 t 3+4 t¿¿

m(t)=−4 t 3+8 t−8¿¿

f. g(z)=(z2+2 z−1¿ ( z−4+35−z2

)

g(z)=(2z+2)( z−4+35−z2

¿+¿(z2+2 z−1¿¿ g.

w(x)=3( x5−2 x+1¿2(5 x4−2)w(x)=(15 x4−6¿¿

h. s(t)=(3t 4-5t 3+ t−1¿5

s(t)=5(3 t 4−5 t 3+t−1¿4 (12 t3−15 t 2+1 )

Luis Requena CI: 23780868s(t)=(60 t 3−75 t2+5¿¿

i. n(y)= 53¿¿

n(y)=−5¿¿n(y)= (−10 y−20 )¿¿n(y)= (−120 y−240 )¿¿

J. f(t)= √2 t+53−t

f(t)=2

2√2t+5(3−t )−√2 t+5 (−1)

1¿¿

f(t)=3−t+2 t+5

√2 t+5¿¿¿

f(t)= 8+t¿¿

k. f(x)=3√ x.4√ x2+4 x+4

f(x)=( 32√ x )(4√ x2+4 x+4)+(3√ x)(4 ¿¿)

f(x)= 6√x2+4 x+4√ x

+ (12 x+24 )√ x√x2+4 x+4

L. y=3senx-5cosx

y= 3cosx +5senx

LL.y=sen3x . cos3x

y=3(cos3x)(cos3x)-(sen3x)(3sen3x)

y=3cos23x-3sen23x

m.y=3sen25x

y=6sen5x cos5x.5y=30sen25x

n. y= tgxsenx−cosx

y= sen2 x (senx−cosx )−tgx (cosx+senx)

¿¿

Luis Requena CI: 23780868O. y= x2 senx

Y=2xsenx+ x2 cosx

Q. y=x√s enx+cosx

y= √senx+cosx +xcosx−xsenx2√senx+cosx

P. y= x2+1xsenx

y= 2x2 senx− (x2+1 )(senx+xcosx)

¿¿

R. f(t)=sen4(t4+3t)

f(t)=4sen3(t4+3t)cos(t4+3t).(4t3+3)

f(t)=(16t3+12)sen3(t4+3t)cos(t4+35)

RR.g(t)=cos2(cos(cost))

g(t)=-2cos(cos(cost)sen(cost(cost)sent(cost)sent

3. Suponiendo que cada una de las siguientes ecuaciones define una

función de x derivable, encuentre y’ o  usando derivación implícita

A.4x2+9y2=36

8x+18yy`=0

Y`=−8 x18 y

=−4 x9 y

B.xy2-x+16=0

y2+x2yy`-1=0

y`− y2+12xy

C.x3-3x2y+19xy=0

Luis Requena CI: 237808683x2-3(2xy+x2y`)+19(y+xy`)=0

3x2-6xy-3x2y`+19y+19xy`=0

y`=−3 x2+6 xy−19 y−3 x2+19 x

D.√ xy+3 y=10 x

y+ xy2√xy +3y`=10

y2√ xy+

xy2√ xy+3y`=10

y`=10− y

2√xyx

2√ xy+3

E.6x-√2xy+x y3= y2

6- 2( y+xy )2√2 xy

+y3+x3y2y`=2yy`

6- y√2 xy+

xy√2 xy+y3+3xy2y`=2yy`

xy√2 xy

+3xy2y`-2yy`=-6+ y√2 xy

-y3

y`=−6 y

√2 xy− y3

x√2 xy

+3 x y2−2 y

F. y2

x3-1= y

32

2 yy x3− y23x2¿¿

-0= 32y12y`

2 yy x3

x6- 3x

2 y2

x6= 32y12y`

Luis Requena CI: 237808682 yyx3

- 3 y2

x4= 32y12y`

2 yx3

- 32y12 y =3 y

2

x4

Y`=

3 y2

x 4

2x3

−32y12

G. xy+seny=x2

y+xy`+y`cosy=2x

y`= 2 x− yx+cosy

H. cos(xy)=y2+2x

-sen(xy).(y+xy`)=2yy`+2

-sen(xy)y-sen(xy)xy`=2yy`+2

-sen(xy)xy`-2yy`=2+sen(xy)y

y`= 2+ ysen(xy )−xsen ( xy )−2 y

I. y= 3x+ y2

x+ y3

xy+y4=3x+y2

y+xy`+4y3y`=3+2yy`

xy`+4y3y`-2yy`=3-y

y`= 3− yx+4 y3−2 y

J. xy+ √x+4x

-5=0

Luis Requena CI: 23780868y+xy`+

1+ y2√ x+ y

x−√x+ y

x2=0

y`=− y− x

2x2√x+ y+ √x+ y

x2

x2 x2√x+ y

+x

y`=− y− 1

2x √x+ y+ √x+ y

x2

12 x√ x+ y

+x

 

4. Sea  una función derivable de x tal que:  . Supóngase que  . hallar  siguiendo estos pasos:

x3y+y4=2

3x2y+x3y`+4y3y`=2

y`= 2−3 x2 y

x3+4 y3

A.x3( 2−3 x2 y

x3+4 y3)+3x2y+4y3( 2−3 x

2 yx3+4 y3

)=0

0=0

B. y`= −3x2 yx3+4 y 3

y`(1)=−3 x2

x3+4