Diferensial Fungsi Majemuk Diferensial Parsial Diferensial Total Chain rule dll

Diferensial Parsial Diferensial Total

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

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

Techniques of partial differentiation Market model Techniques of partial differentiation

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

Market model

Q S D P D1 Q S1 D P S0

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

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

Input-output model ∂x1/∂d1 = b11

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

Total Differentials

Diferensial Total

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

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

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)

z x y Kaidah Rantai t Pohon rantai

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

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