lect12_5

lect12_5 - Chain Rule - (12.5) 1. Chain Rule Let z ! f!x,...

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Chain Rule - (12.5) 1. Chain Rule Let z ! f ! x , y " and x ! x ! t " and y ! y ! t " . So implicitly z is a function of t , z ! g ! t " . What is dz dt ? lim h " 0 g ! t # h " ! g ! t " h ! lim h " 0 f ! x ! t # h " , y ! t # h "" ! f ! x ! t " , y ! t "" h ! lim h " 0 f ! x ! t # h " , y ! t # h "" ! z ! x ! t # h " , y ! t "" # f ! x ! t # h " , y ! t "" ! z ! x ! t " , y ! t "" h ! lim h " 0 f ! x ! t # h " , y ! t # h "" ! f ! x ! t # h " , y ! t "" h # lim h " 0 f ! x ! t # h " , y ! t "" ! f ! x ! t " , y ! t "" h By the Mean Value Theorem, we know there exist x 1 between x ! t # h " and x ! t " ; and y 1 between y ! t # h " and y ! t " such that f ! x ! t # h " , y ! t # h "" ! f ! x ! t # h " , y " ! f y ! x ! t # h " , y 1 "! y ! t # h " ! y ! t "" f ! x ! t # h " , y ! t "" ! f ! x ! t " , y ! t "" ! f x x 1 , y ! t " ! x ! t # h " ! x ! t "" So lim h " 0 g ! t # h " ! g ! t " h ! lim h " 0 f y ! x
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This note was uploaded on 02/05/2012 for the course MATH 2142 taught by Professor Lerna during the Fall '10 term at FIU.

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lect12_5 - Chain Rule - (12.5) 1. Chain Rule Let z ! f!x,...

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