This document discusses implicit differentiation and provides examples of taking the derivative of implicit functions. It begins by presenting the general form of an implicit function as f(x,y)=0 and provides some examples. It then shows how to take the derivative dy/dx of several implicit functions by applying the chain rule and implicit differentiation. Formulas for the derivatives of inverse trigonometric functions are also provided, along with examples of finding dy/dx for functions involving inverse trigonometric functions.

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

2. Bentuk Umum:
f (x,y) = 0
Contoh:
1.x2
+ y3
= 0
2.x3
+ 5xy + y4
+3 = 0
3.2x4
– 3y +5= 2y2
4.dll

3. Tentukanlah dy/dx dari fungsi implisit berikut ini:
1. x3
+ y4
= 0
2. x5
+ xy + y3
+4 = 0
3. 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

4. 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
)

5. 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)

6. 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’ = -
2
x1
1
−
2
x1
1
−
2
1
1
x+
2
1
1
x+

7. Contoh 1:
Tentukanlah dy/dx dari fungsi berikut:
1. y = arc sin (5 + x2
)
2. y = arc tg (5x/9)

8. 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 .
=
2
1
1
u−
22
)5(1
1
x+−
22
)5(1
1
x+−
24104
2
24
−−− xx
x

9. 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
5
1
1






+
x
2
9
5
1
1






+
x
2
9
5
1
9/5






+
x

10. TERIMA KASIH
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