Math-review0 - ECON 302 Math Review Ali Toossi Marginal analysis Concept of Derivative its application in Economics Suppose Y = f(X Define Y = f(X

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ECON 302 Math Review Ali Toossi Suppose Y = f (X). Define ΔY = f (X + ΔX) – f (X), where ΔY and ΔX denote changes in Y and X respectively. Slope at X 0 = Derivative of the function at X 0 : X Y dX dY X = 0 lim Example1: Consider a linear function Y = f (X) = a + m X. Find slope at X = X 0 . Solution : ΔY = f (X 0 + ΔX) – f (X 0 ) = [a + m (X 0 + ΔX)] – [a + m X 0 ] = m ΔX Slope at X 0 = Derivative of the function at X 0 = X Y dX dY X = 0 lim = 0 lim X (m X) / X = 0 lim X m = m. So slope is constant and does not depend on X 0 . Example2: Consider the function Y = f (X) = a X 2 . Find slope at X = X 0 . Solution : ΔY = f (X 0 + ΔX) – f (X 0 ) = a (X 0 + ΔX) 2 – [a X 0 2 ] = (a X 0 2 + 2a X 0 ΔX + a ΔX 2 ) - a X 0 2 = 2a X 0 ΔX + a ΔX 2 Y/ X = (2a X 0 ΔX + a ΔX 2 ) / X = 2a X 0 + a ΔX Slope : X Y X 0 lim = 0 lim X ( 2a X 0 + a ΔX) = 2a X 0 . So slope is not constant and depends on X 0 . For example if X 0 = 4, slope is 8a , but if X 0 = 0.5, slope is a . Rules of Differentiation (i.e. rules of finding the derivatives of functions): Constant-Function Rule 0 = = dX dY K Y where k is a constant Product Rule dx x dg x f dx x df x g dX dY x g x f Y ) ( ) ( ) ( ) ( ) ( ) ( + = = Power Function Rule 1 - = = n n ncx dX dY cx Y , where c and n are constants. Quotient Rule ) ( ) ( ) ( ) ( ) ( ) ( ) ( 2 x g dx x dg x f dx x df x g dX dY x g x f Y - = = Sum-Difference Rule dx x dg dx x df dX dY x g x f Y ) ( ) ( ) ( ) ( ± = ± = Chain Rule Suppose that we have a function Z = f(Y) where Y = g(X), then: dx dy dy dz dx dz = 1
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Partial Derivatives: Suppose Y = f(K, L), i.e. Y depends on K & L both. We are interested in knowing what happens to Y if we change K by an infinitesimally small amount. That is, suppose we want to know the derivative of Y with respect to K. What about L? Suppose we are prepared to assume that changing K by a little bit does not cause L to change at all . In such a case we can find the partial derivative of Y with respect to K. The partial derivative is defined as the effect on Y of a very small change in K holding L constant . The partial derivative of Y with respect to K is written as:
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This note was uploaded on 02/14/2011 for the course ECON 302 taught by Professor Avrin-rad during the Spring '09 term at University of Illinois, Urbana Champaign.

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Math-review0 - ECON 302 Math Review Ali Toossi Marginal analysis Concept of Derivative its application in Economics Suppose Y = f(X Define Y = f(X

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