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# Class Note for EconS 521 at WSU

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## About this Document

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This 2 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at Washington State University taught by a professor in Fall. Since its upload, it has received 17 views.

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Date Created: 02/06/15
EconS 521 Fall 2008 l Liebnitz Rule for Differentiating an Integral Let the functions f R2 gt R a R gt R and b R gt R be continuously differentiable and define the integral lnction Fy jiffy fxydx 1 Then the derivative of F with respect to y satisfies the following property My 6 F U M6196 f My yb39y f 00 ya39y 2 ay Proof Let A at 0 denote an arbitrary change in y We then have byA F y A WW fx y Adx a A a A b A 1 yl fxyAdx J W fxyAdxJ W fxyAdx ay ay ayA b A A 3 y a y j fxyAdx j fxyAdx a y 110 b b A a A I y fx y Adx j W fx y Adx j W fx y Adx 110 MY 40 Therefore subtracting 1 from the last line of 3 and dividing the result by A gives m f lim Agt0 A Aao A ay j bltyAfxyAdx j M Jaw mdx lim My 1im 1 4 Agt0 A Aao A My fxyAdx J a x y dx lim My 1im W am By Agt0 A Agt0 A b a J OHN J yAfxyAdx EconS 521 Fall 2008 2 We can then rewrite the second term in the bottom line of 4 as byA fx y Adx byA lim39ibm zlimj lim fxayA fxy dx A90 A Agt0 My A90 A byA l39 by fx ydx A EE A 5 M gt zl y Mawwowww My 6y f byyb39y The first equality follows from the fact that the limit of the limit is the limit The first term in the second equality follows from the de nition of the partial derivative of f with respect to y and the second term in the second equality follows from the fundamental theorem of calculus 7 ie differentiation and integration are inverse functions The bottom line of 5 follows from the fact that there is zero area under a point ie in an integral that has the same upper and lower limit Similarly we can rewrite the integral and limit on the third line of 4 as ayA fx y Adx ayA 100 fxay A fx Agt0 A Agt0 Agt0 A a y ayA 1 gm fxydx 33 A 6 a zl y Mabw awMWU am By f My ya39y Therefore combining 446 we obtain Liebnitz Rule as stated in 2 above I

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