pt 2 turunan fungsi eksponen, logaritma, implisit dan cyclometri-d4
TRANSCRIPT
MATEMATIKA
Oleh:Dr. Parulian Silalahi, M.Pd
http://polmansem3.esy.es/
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
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
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 +
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 +
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)
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)
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
u−22 )5(1
1
x+−
22 )5(1
1
x+−
2410
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