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Unformatted text preview: Univariate Calculus Professor Erkut Ozbay Economics 300 Outline Rules of differentiation Sum rule Scale rule Product rule Power rule Exponential rule Chain rule Quotient rule Logarithmic rule Outline Second derivatives and convexity Economic applications Risk aversion and utility functions Elasticities Total revenue Average, marginal, and total cost Rules of differentiation Used to evaluate derivative of any function Much easier than working with basic definition ( ) ( ) lim x dy f x x f x dx x +  = Sum rule ( ) ( ) ( ) ( ) ( ) ( ) y h x f x g x dy h x f x g x dx = = + = = + Scale rule ( ) ( ) ( ) ( ) y h x kf x dy h x kf x dx = = = = Product rule ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) y h x f x g x dy h x f x g x f x g x dx = = = = + Power rule 1 ( ) ( ) n n y h x kx dy h x nkx dx = = = = Exponential rule ( ) ( ) kx kx y h x e dy h x ke dx = = = = Chain rule ( ) ( ( )) ( ) ( ( )) ( ) y h x g f x dy h x g f x f x dx = = = = ( ) ( ) y g u u f x dy dy du dx du dx = = = Quotient rule 2 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) f x y h x g x dy f x g x f x g x h x dx g x = = = = Logarithmic rule ( ) ln 1 ( ) ( ) log 1 1 ( ) ln b y f x x dy f x dx x y f x x dy f x dx x b = = = = = = = = Exercise 3 4 2 2 25 7 4 8 3 14 1 2 3 y x y y x y x x y x x = = = = + = 3 2 dy x dx = 5 7 dy x dx =  1/2 3 16 2 dy x x dx = + 3 2 2 3 dy x dx =  Exercise 2 2 2 4 20 4 3ln 2( 1) x y x x y e y x y x = + = = = + 20 8 dy x dx = 2 2 x dy e dx = 3 dy dx x = 4( 1) dy x dx = + 2...
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 Fall '08
 cramton
 Economics

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