Nama : Siti Fatimah

NPM : 0031427

Kelas : 1 EA

Mata Kuliah : Matematika 2

TUGAS 2

Tentukan dx/dy dari:

1 . y= √ x5+6x2+3

2 . y=3√ x4+6 x+1

3 . y=5√ x2−5 x

4 . y=1

√x4+2x

5 . y=1

3√x2−6 x

6 . y=1

5√x2−5 x+2

7 . y=sin√ x2+6 x

8 . y=cos3√x3+2

9 . y=sin1

√x2+2

10 . y=cos1

3√ x2+6Penyelesaian:

1 . y= √ x5+6x2+3 misal u =x5+6 x2+3 , du/dx =5 x4+12 x

y=√u= u12 , dy/du =1

2u

−12 =12(x5+6 x2+3)

−12

dy/dx = du/dx . dy/du

= (5 x4+12 x).12(x5+6 x2+3)

−12

=12

.(5 x4+12 x).(x5+6 x2+3)−12

2 . y=3√ x4+6 x+1 misal u =x4+6 x+1 , du/dx =4 x3+6

y=3√u= u13 , dy/du =1

3u

−23 = 1

3( x4+6 x+1)

−23

dy/dx = du/dx . dy/du

= (4 x3+6).13( x4+6 x+1)

−23

=13

.(4 x3+6).(x4+6 x+1)−23

3 . y=5√ x2−5 x

misal u =x2−5 x , du/dx =2 x−5

y=5√u= u15 , dy/du =1

5u

−45 = 1

5( x2−5x )

−45

dy/dx = du/dx . dy/du

= (2 x−5).15( x2−5x )

−45

=15

.(2 x−5).(x2−5 x)−45

4 . y=1

√x4+2x misal u =x4+2 x , du/dx =4 x3+2

misal v=√u= u12 , dv/du =1

2u

−12 =

12¿

y=1v=v−1 , dy/dv = −1v−2 = −1(√ x4+2 x)−2

dy/dx = du/dx . dv/du . dy/dv

= (4 x3+2).12(x4+2x )

−12 .−1(√ x4+2 x)−2

=−12

.(4 x3+2).(x4+2 x)−12 .(√ x4+2 x)−2

5 . y=1

3√x2−6 x

misal u =x2−6 x , du/dx =2 x−6

misal v=3√u= u13 , dv/du =1

3u

−23 =

13¿

y=1v=v−1 , dy/dv = −1v−2 = −1(

3√ x2−6 x)−2

dy/dx = du/dx . dv/du . dy/dv

= (2 x−6).13¿.−1(

3√ x2−6 x)−2

=−13

.(2 x−6).(x2−6 x )−23 .(

3√x2−6 x )−2

6 . y=1

5√x2−5 x+2

misal u =x2−5 x+2, du/dx =2 x−5

misal v =5√u= u15 , dv/du =1

5u

−45 =

15¿

y=1v=v−1 , dy/dv = −1v−2 = −1(

5√ x2−5 x+2)−2

dy/dx = du/dx . dv/du . dy/dv

= (2 x−5).15¿.−1(

5√ x2−5 x+2)−2

=−15

.(2 x−5).(x2−5 x+2)−45 .(

5√x2−5 x+2)−2

7 . y=sin√ x2+6 x misal u =x2+6 x , du/dx =2x+6

misal v =√u = u12 , dv/du=1

2u

−12 =12(x¿¿2+6 x )

−12 ¿

y = sin v , dy/dv =cos v= cos √u = cos √x2+6 x dy/dx = du/dx . dv/du . dy/dv

=¿2x+6¿.12(x¿¿2+6 x )

−12 ¿ .cos √x2+6 x

=12¿2x+6¿.(x¿¿2+6 x)

−12 .cos√ x2+6 x ¿

8 . y=cos3√x3+2

misal u =x3+2 , du/dx =3 x2

misal v =3√u= u13 , dv/du=1

3u

−23 =13( x¿¿3+2)

−23 ¿

y = cos v , dy/dv =−sin v= −sin 3√u = −sin 3√ x3+2 dy/dx = du/dx . dv/du . dy/dv

=(3 x2).13( x¿¿3+2)

−23 ¿ .−sin 3√ x3+2

=−13

(3 x2).¿¿

9 . y=sin1

√x2+2

y=sin ¿

misal u = x2+2 , du/dx = 2x

misal v =√u = u12 , dv/du = 1

2u

−12 = = 1

2(x¿¿2+2)

−12 ¿

y=sin v−1 , dy/dv = cos v−1= cos (√u)−1= cos (√ x2+2)−1

dy/dx = du/dx . dv/du . dy/dv

=(2 x).12(x¿¿2+2)

−12 ¿ . cos (√ x2+2)−1

=x (x¿¿2+2)−12 ¿ . cos (√ x2+2)−1

= x (x¿¿2+2)−12 ¿ . cos

1

√ x2+2

10 . y=cos1

3√ x2+6 y=cos ¿

misal u = x2+6 , du/dx = 2x

misal v =3√u = u13 , dv/du = 1

3u

−23 = 1

3( x¿¿2+6)

−23 ¿

y=cos v−1 , dy/dv = −sin v−1= −sin( 3√u)−1= −sin(3√x2+6)−1

dy/dx = du/dx . dv/du . dy/dv

=(2 x).13( x¿¿2+6)

−23 ¿ . −sin(

3√x2+6)−1

= −13

(2x ).(x¿¿2+6)−23 ¿ . sin(

3√x2+6)−1

= −13

(2x ).(x¿¿2+6)−23 ¿ . sin

13√x2+6