2DerivativeandRulesofDifferentiation.docx

1、2DerivativeandRulesofDifferentiationSimple DerivativesEconomic Theory Relationships between variables functionse.g. Price depends on quantity sold Cost depends on quantity sold Employment depends on investment Cost is a function of quantity(For every value of quantity there exists one and only one v

2、alue of cost) C = f(Q), C = (Q) or even C = C(Q) Specifically:May be written asC = 10 +3QQ = 1, 2, 3, 4, . . . 13, 16, 19, 22, . . . Note C = 15, i.e.C is still (mathematically) a function of quantity because it satisfied the definitionQ = 1, 2, 3, 4, . . . 15, 15, 15, 15, . . .DerivativesA common p

3、roblem is this: If C = f(Q), How does C change when Q changes?i.e. What is the rate of change of C with regard to (WRT) Q?The answer is given by the derivative of C WRT Q.The derivative is written or CIn economics the derivative is equivalent to a marginal quantity (Marginal cost, Marginal Revenue,

4、Marginal Product)In general Y = f(X)Where: Y is the dependent variable X is the independent variableDerivative or YIt is (a) the rate of change of Y WRT X or(b) by how much does Y change when X changes by one (very small) unit?or(c) on a graph the slope (same as (b)Slope is = Rules for DerivativesPo

5、wer Function RuleD1 If Y = a Xb a,b = constants, numbersY = = b a Xb-1e.g.C = 3Q2C = 2*3Q2-1 = 6Q1= 6QThe rule can be applied to function with several terms.e.g. C = 10+4Q3 + 6QC = = 12Q2 + 3QDerivative of a constant (10)=0C = 10 + 3QC = 3 Q1-1= 3Q0= 3Product RuleSay C = (3+2Q)(1+Q2) C=?D2 If U and

6、V are functions of Xe.g. C = (3+2Q)(1+Q2)C = (3+2Q)2Q + (1+Q2)2 = 6Q + 4Q2 + 2 + 2Q2= 2 +6Q+6Q2Quotient RuleD3 If U and V are functions of Xe.g. C = C = = 2Chain RuleSay C = 3 +10Q Q = 20-2PPQC: change P change in Q change in CWhat is? How can we find?D4 X = f(Y) Y = f(Z)e.g.C = 3 +10QQ = 20-2PC = =

7、 (10) * (-2) = -20e.g.Y = Y = ?Y = Y = f(Z)Z = F(X) = = Long chains:Say W = f(Y)Y = g(X)X= h(Z)etc.Inverse RuleSometimes we have but we want. The rule is simpleD5 e.g.(i) C = 10 + 3Q= ?= 3= (ii) C = 10 + 3Q + 0.1 Q2 = ?= 3 + 0.2Q= Exponential Function e is a number = 2.71828 Consider eX. X is the Po

8、wer (This is not like Xa)D6 (The derivative has the same value as eX)Illustration:X = 0123eX =12.77.420.1 = 12.77.420.1(Note : these figures are to one decimal place)i.e. When X = 2; ex = 2.7 and one small unit change in X will cause ex to increase by 2.7 of those units. So as X gets bigger, eX incr

9、eases faster. It speeds up.Consider eaX (a = a constant or number)D7 (Not the same as)e.g. Now consider loge X (also written log X or ln X or the natural log of X)D8 e.g. Y = 3 log X The rate of change of log X gets smaller as X gets bigger (It slows down)We can prove this if we assume X = eYLog X = YY = log XThen D5