MATEMATIKA II

Oleh:Dr. Parulian Silalahi, M.Pd

http://polmansem3.esy.es/

Bentuk Umum:

f (x,y) = 0

Contoh:

1.x2 + y3 = 0

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

3.2x4 – 3y +5= 2y2

4.dll

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

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)

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)

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

Contoh 1:

Tentukanlah dy/dx dari fungsi berikut:

 

1. y = arc sin (5 + x2)

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

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

u22 )5(1

1

x

22 )5(1

1

x

24104

224 xx

x

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

TERIMA KASIHSelamat Belajar