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Chap14_Sec5

# Chap14_Sec5 - PARTIAL DERIVATIVES 14.5 The Chain Rule In...

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14.5 The Chain Rule PARTIAL DERIVATIVES In this section, we will learn about: The Chain Rule and its application in implicit differentiation.

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Suppose that z = f ( x , y ) is a differentiable function of x and y (i.e., f x and f y are continuous), where x = g ( t ) and y = h ( t ) are both differentiable functions of t . square4 Then, z is a differentiable function of t and Suppose z = f ( x , y ) is a differentiable function of x and y , where x = g ( s , t ) THE CHAIN RULE (CASE 1) dz f dx f dy dt x dt y dt = + THE CHAIN RULE (CASE 2) and y = h ( s , t ) are differentiable functions of s and t . square4 Then, = + = + z z x z y z z x z y s x s y s t x t y t TREE DIAGRAM
Suppose u is a differentiable function of the n variables x 1 , x 2 , …, x n and each x j is a differentiable function of the m variables t 1 , t 2 . . . , t m . Then, u is a function of t 1 , t 2 , . . . , t m THE CHAIN RULE (GEN. VERSION) and for each i = 1, 2, . . . , m.

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