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14.5 - 14.5 THE CHAIN RULE KIAM HEONG KWA We shall assume...

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14.5: THE CHAIN RULE KIAM HEONG KWA We shall assume all functions are differentiable in this section. 1. The Chain Rule Suppose that f is a function of the n variables x 1 , x 2 , · · · , x n and each x i is a function of the m variables t 1 , t 2 , c · · · , t m , then (1.1) ∂f ∂t j = n X i =1 ∂f ∂x i ∂x i ∂t j for each j , j = 1 , 2 , · · · , m . It should be noted that ∂f/∂t j is evaluated at ( t 1 , t 2 , · · · , t m ), ∂f/∂x i is evaluated at the corresponding value of ( x 1 , x 2 , · · · , x n ), i.e. at ( x 1 ( t 1 , t 2 , · · · , t m ) , x 2 ( t 1 , t 2 , · · · , t m ) , · · · , x n ( t 1 , t 2 , · · · , t m )) , and ∂x i /∂t j is evaluated at ( t 1 , t 2 , · · · , t m ). For instance, if f is a function of x , y , and z , while x , y , and z are functions of t , then df dt = ∂f ∂x dx dt + ∂f ∂y dy dt + ∂f ∂z dz dt . The functional dependence of these variables can be denoted by the tree diagram f x t y t z t As another instance, if f is a function of x , y , and z , while x , y , and z are functions of r , s , and t
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