Diferensial Fungsi Majemuk

Presentasi berjudul: "Diferensial Fungsi Majemuk"— Transcript presentasi:

1 Diferensial Fungsi Majemuk
Diferensial Parsial Diferensial Total Chain rule dll

2 Diferensial Parsial Diferensial Total

3 High Order Partial Derivatives
Fungsi dengan lebih dari satu variabel bebas juga dapat diturunkan lebih dari satu kali Turunan parsial z = f (x,y)  kalau kontinyu dapat mempunyai turunannya sendiri.  empat turunan parsial : Dapat dilambangkan fxx, fxy, fyx, dan fyy fxy = fyx

4 Partial derivatives Cobb-Douglas production function (+=1)
Q = 96K0.3 L0.7

5 Techniques of partial differentiation
Market model Techniques of partial differentiation

6 Geometric interpretation of partial derivatives
Market model Geometric interpretation of partial derivatives

7 Market model

8 Q S D P D1 Q S1 D P S0

9 Q S0 D P S1 Q S0 D1 D0 P Q0 Q1 Market model

10 National-income model
Y = C + I0 + G0 C = a + b(Y-T); b = MPC (a > 0; 0 < b < 1) T=d+tY; t = MPT (d > 0; 0 < t < 1) Y=( a-bd+I+G)/(1-b+tb) C=(b(1-t)(I+G)+a-bd)/ (1-b+tb) T=(t(I+G)+ta+d(1-b))/ (1-b+tb) National-income model

11 Input-output model ∂x1/∂d1 = b11

12 Note on Jacobian Determinants
Use Jacobian determinants to test the existence of functional dependence between the functions /J/ Not limited to linear functions as /A/ (special case of /J/ If /J/ = 0 then the non-linear or linear functions are dependent and a solution does not exist. Note on Jacobian Determinants

13 Total Differentials

14 Diferensial Total

15 Let Utility function U = U (x1, x2, …, xn)
Differentiation of U wrt x1..n U/ xi is the marginal utility of the good xi dxi is the change in consumption of good xi

16 Finding the total derivative from the differential
Given a function y = f (x1, x2, …, xn) Total differential dy is: Total derivative of y with respect to x2 found by dividing both sides by dx2 (partial total derivative) Finding the total derivative from the differential

17 Chain rule (kaidah rantai)
This is a case of two or more differentiable functions, in which each has a distinct independent variable. where z = f(g(x)), i.e., z = f(y), i.e., z is a function of variable y and y = g(x), i.e., y is a function of variable x If R = f(Q) and if Q = g(L)

18 z x y Kaidah Rantai t Pohon rantai

19 Kaidah Rantai Kalau w = w(x,y,z) dan x = x(u,v), y = y(u,v), dan z = z(u,v), maka pohon rantai : w y v z u x

20 Kalau z = z(x,y), dan x = x(s), y = y(s), dan s = s(u,v), maka pohon rantai menjadi :