# Sec12.4 - Sec.12.4 The Chain Rule The Chain Rule case 1: If...

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Sec.12.4 The Chain Rule The Chain Rule case 1: If z = f(x,y) is a differentiable function of x and y , where x = g(t) and y = h(t) are both differentiable functions of t , then z is a differentiable function of t and dz z dx z dy dt x dt y dt      . Note that this can be extended. EX 1 Use the Chain Rule to find dz dt for 23 2 sin( ), , tt zx y x e y e  . The Chain Rule case 2: If z = f(x,y) is a differentiable function of x and y , where x = g(s,t) and y = h(s,t) are both differentiable functions of s and t , then z is a differentiable function of s and t and , zz x z y x z y sx s y s tx t y t . Note that this can be extended. The easiest way to keep the chain rule straight is to use a tree diagram. EX 2 Use the Chain Rule to find and ww st for  2 3,l n , s wx y x s t y t  .

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Formula for Implicit Differentiation: x y F dy dx F where F(x,y) = 0 and y is a differentiable function of x. EX 3 Find dy dx for 32 (, ) 3 0 . Fxy yx xy  Can be extended: , F F y zz x FF   
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## This note was uploaded on 02/03/2010 for the course MATH 2224 taught by Professor Mecothren during the Spring '03 term at Virginia Tech.

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Sec12.4 - Sec.12.4 The Chain Rule The Chain Rule case 1: If...

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