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MATEMATIKA Oleh: Dr. Parulian Silalahi, M.Pd http://polmansem3.esy.es/

Pt 2 turunan fungsi eksponen, logaritma, implisit dan cyclometri-d4

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Page 1: Pt 2 turunan fungsi eksponen, logaritma, implisit dan cyclometri-d4

MATEMATIKA

Oleh:Dr. Parulian Silalahi, M.Pd

http://polmansem3.esy.es/

Page 2: Pt 2 turunan fungsi eksponen, logaritma, implisit dan cyclometri-d4

Rumus Dasar:

y = ex y’ = ex

y = e-x y’ = - e-x

y = eax y’ = a. eax

y = e-ax y’ = -a e-ax

Contoh 5:

Tentukan dy/dx dari fungsi hiperbolik berikut:

1. y = ecos5x

2. y = (e4x – e5x)4

Page 3: Pt 2 turunan fungsi eksponen, logaritma, implisit dan cyclometri-d4

Jawab:1. y = ecos5x

mis u = cos 5x du/dx = - 5.sin 5xy = eu dy/du = eu = ecos5x

dy/dx = du/dx . dy/du = - 5.sin 5x. ecos5x

2. y = (e4x – e5x)4

mis u = (e4x – e5x) du/dx = 4e4x –5 e5x

y = u4 dy/du = 4u3 = 4(e4x – e5x)3

dy/dx = du/dx . dy/du = (4e4x –5 e5x). 4(e4x – e5x)3

= 4(4e4x –5 e5x). (e4x – e5x)3

Page 4: Pt 2 turunan fungsi eksponen, logaritma, implisit dan cyclometri-d4

Rumus Dasar:

1. y = alog x y’ =1/a. alog e

2. y = ln x y’ = 1/x elog e =1 /x

3. y = ax y’ = ax. ln a

Contoh 2.6:

Tentukan dy/dx dari fungsi logaritma berikut:

1. y = ln (x2 + 5)

2. y = )6( 2

3 xx +

Page 5: Pt 2 turunan fungsi eksponen, logaritma, implisit dan cyclometri-d4

Jawab:

1. y = ln (x2 + 5)

mis: u = x2 + 5 du/dx = 2x

y = ln u dy/du = 1/u = 1/(x2 + 5)

dy/dx = du/dx . dy/du = 2x . 1/(x2 + 5) = 2x/(x2 + 5)

2. y =

mis: u = x2 + 6x du/dx = 2x + 6

y = 3u dy/du = 3u . ln 3 = . ln 3

dy/dx = du/dx . dy/du = (2x + 6) . ln 3

)6( 2

3 xx +

)6( 2

3 xx +

)6( 2

3 xx +

Page 6: Pt 2 turunan fungsi eksponen, logaritma, implisit dan cyclometri-d4

Rumus Dasar:

1. y = sinh x y’ = cosh x

2. y = cosh x y’ = sinh x

Contoh 7:

Tentukan dy/dx dari fungsi hiperbolik berikut:

1. y = sinh 7x

2. y = cosh3 (1-x)

Page 7: Pt 2 turunan fungsi eksponen, logaritma, implisit dan cyclometri-d4

Jawab:1. y = sinh 7x mis u = 7x du/dx =7y = sinh u dy/du = cosh u = cosh 7xdy/dx = du/dx . dy/du = 7. cosh 7x2. y = cosh3 (1-x)mis u = 1 – x du/dx = -1t = cosh u dt/du = sinh u = sinh (1-x)y = t3 dy/dt = 3 t2 = 3 cosh2 (1-x)dy/dx = du/dx . dt/du. dy/dt = -1. sinh (1-x). 3 cosh2 (1-x)

= -3 sinh (1-x). cosh2 (1-x)

Page 8: Pt 2 turunan fungsi eksponen, logaritma, implisit dan cyclometri-d4

Bentuk Umum:

f (x,y) = 0

Contoh:

1.x2 + y3 = 0

2.x3 + 5xy + y4 +3 = 0

3.2x4 – 3y +5= 2y2

4.dll

Page 9: Pt 2 turunan fungsi eksponen, logaritma, implisit dan cyclometri-d4

Tentukanlah dy/dx dari fungsi implisit berikut ini:

1. x3+ y4 = 02. x5+ xy + y3 +4 = 03. x4 – 3y +5xy= 4y2

Jawab:

1. x3+ y4 = 0

d/dx (x3+ y4 )= d/dx (0)

3x2 dx/dx + 4y3 dy/dx = 0

3x2 + 4y3 dy/dx = 0

dy/dx = -3x2 / 4y3

Page 10: Pt 2 turunan fungsi eksponen, logaritma, implisit dan cyclometri-d4

2. x5+ xy + y3 +4 = 0

d/dx(x5+ xy + y3 +4 )= d/dx (0)

5x4 dx/dx + 1 dx/dx. y +x. dy/dx + 3y2 dy/dx + 0 = 0

5x4 + y +x. dy/dx + 3y2 dy/dx + 0 = 0

x. dy/dx + 3y2 dy/dx = - 5x4 - y

(x + 3y2) dy/dx = - (5x4 + y)

dy/dx = -(5x4+ y)/(x + 3y2)

Page 11: Pt 2 turunan fungsi eksponen, logaritma, implisit dan cyclometri-d4

3. x4 – 3y +5xy= 4y2

d/dx(x4 – 3y +5xy) = d/dx (4y2)

4x3 dx/dx – 3 dy/dx + 5 dx/dx. y + 5x dy/dx = 8y dy/dx

4x3 – 3 dy/dx + 5y + 5x dy/dx = 8y dy/dx

-3 dy/dx + 5x dy/dx – 8y dy/dx = - 4x3 – 5y

( -3 + 5x – 8y) dy/dx = - (4x3 + 5y)

dy/dx = -( 4x3 + 5y)/(-3 + 5x – 8y)

Page 12: Pt 2 turunan fungsi eksponen, logaritma, implisit dan cyclometri-d4

Rumus Dasar

1.y = arc sin x y’ =

2. y = arc cos x y’ = -

3. y = arc tg x y’ =

4. y = arc cot x y’ = -

2x1

1

2x1

1

21

1

x+

21

1

x+

Page 13: Pt 2 turunan fungsi eksponen, logaritma, implisit dan cyclometri-d4

Contoh 1:

Tentukanlah dy/dx dari fungsi berikut:

1. y = arc sin (5 + x2)

2. y = arc tg (5x/9)

Page 14: Pt 2 turunan fungsi eksponen, logaritma, implisit dan cyclometri-d4

Jawab:

1. y = arc sin (5 + x2)

misalkan: u = 5 + x2 du/dx = 2x

y = arc sin u dy/du = =

dy/dx = du/dx . dy/du = 2x . =

21

1

u−22 )5(1

1

x+−

22 )5(1

1

x+−

2410

224 −−− xx

x

Page 15: Pt 2 turunan fungsi eksponen, logaritma, implisit dan cyclometri-d4

2. y = arc tg (5x/9)

misalkan: u = 5x/9 du/dx = 5/9

y = arc tg u dy/du =

dy/dx = du/dx . dy/du = 5/9 .

=

2

9

51

1

+ x

2

9

51

1

+ x

2

9

51

9/5

+ x

Page 16: Pt 2 turunan fungsi eksponen, logaritma, implisit dan cyclometri-d4

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