Chiang_Ch7 - Chiang,Ch.7RulesofDifferentiationand

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1 Chiang, Ch. 7   Rules of Differentiation and  Their Use in Comparative Statics • 7.1 Rules of Differentiation for a Function of One  Variable • 7.2 Rules of Differentiation Involving Two or More  Functions of the Same Variable • 7.3 Rules of Differentiation Involving Functions of  Different Variables • 7.4 Partial Differentiation • 7.5 Applications to Comparative-Static Analysis • 7.6 Note on Jacobian Determinants
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2 7.1 Rules of Differentiation  for a Function of One Variable • 7.1.1 Constant-Function Rule • 7.1.2 Power-Function Rule • 7.1.3 Power-Function Rule Generalized
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3 A Review of the 9 Rules of Differentiation  for a Function of One Variable ( 29 ( 29 [ ] ( 29 ( 29 difference - sum function power function contant 0 1 x g x f x g x f dx d nx x dx d k dx d n n ± = ± = = -
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4 Difference Quotient f (x 0 + x) f (x) f (x 0 ) x 0 x 0 + x y= f (x) x Secant slope: rise/run = f (x 0 + x)- f (x 0 )/ (x 0 + x-x 0 )
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5 Difference Quotient f (x) f (x) f (N) N x y= f (x) x Secant slope: rise/run = f (x)- f (N)/ (x-N)
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6 6.2 Rate of Change and the Derivative ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 N x N f x f N f dx dy x N x x N x x N x x x x x f x x f x f dx dy N x x - - = = = = - = - = - - = = lim ) 6 such that 0 ) 5 ) 4 ) 3 ) 2 Let lim ) 1 0 0 0 0 0 0
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7 7.1.1 Constant-Function Rule ( 29 output in change a w/ change t don' cost Fixed 0 dTFC/dQ then c, TFC if costs, fixed total is TFC Example 0 0 lim lim then If ) ( ) ( lim 0 x of values all for zero is function constant a of derivative •The = = = = - - = = - - = = = = = = N x N x N x N x k k k f(N) k f(x) N x N f x f f '(N) dx dy k dx d dx dy k x f y
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8 7.1.1 Power-Function Rule ( 29 ( 29 ( 29 1 1 lim lim 1 n let lim then if lim 1 1 1 1 1 1 1 = = - - = - - = = = - - = = = = = = - - N x N x n n N x n n N x n n N x N x N x N x f '(N) N f(N) x f(x) N x N f x f f '(N) x x dx d nx x dx d x x f y
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9 ( 29 ( 29 ( 29 N x N x N xN x N x N x N x N x N x N x N x f '(N) x x dx d nx x dx d x x f y N x N x n n N x n n + - - - - - = + = - - = - - = = = = = - - ) ( x 2 lim lim 2 n let lim 2 function power a of derivative The 2 2 2 2 2 1 2 2 1 2
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10 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 N x N x N N xN x N x N N x x N x N x N xN x N x N x N x N x f '(N) x x dx d nx x dx d x x f y N x N x n n N x n n - + = + + - - - - - = + + = - - = - - = = = = = - - 2 2 2 2 2 3 3 3 2 2 2 3 3 1 3 3 1 3 x 3 lim lim 3 n let lim 3 function power a of derivative The
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11 7.1.1 Power-Function Rule ( 29 ( 29 ( 29 0 0 lim 1 1 lim 0 n let lim then if lim 0 0 1 0 0 1 0 = = - - = - - = = = - - = = = = = = - - N x N x n n N x n n N x n n N x N x N x f '(N) N f(N) x f(x) N x N f x f f '(N) x x dx d nx x dx d x x f y
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12 7.1.2 Power-Function Rule 3 4 1 1 4x dy/dx then , x y If Example then , If equals n function x the of derivative The ) 2 = = = = = - n- n n n nx f '(x) x f(x) nx x dx d
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Chiang_Ch7 - Chiang,Ch.7RulesofDifferentiationand

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