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03.27.08_Page_1

# 03.27.08_Page_1 - More on the Chain Rule(s Directional...

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Unformatted text preview: More on the Chain Rule(s) Directional Derivatives Chain rule in one variable Chain rule in two variables (v1) x = g0) y = W) Z = fOW) = f(g(t)=h(t)) dz _ 52 dz 5x dy __+__ 3— 5x dt 5y dt Chain rule in two variables (v2) x = g(5, t) y = g0, I) Z = f(x,y) = f(g(Saf): M570) Assume ﬁg,h are differentiable functions 52_5z 52 52 5y __+__ 5—6x 5s 6y 53 52_ 52 5x 52 5y ____+__ 5: 5x5t 5y 5t Chain rule v2 follow from chain rule v1 52 d 5(S07t0)=g 5:30 f(g(50,t0)h(30,t0)) 6f 62' :—— 7 7f 62 d8 Sisng(S0 0) +ﬂi WM) 5y d5 “ 6f 5g 5f (572 =—— s ,t +—— s ,r 6x 5,8(0 0) 5y53(0 0) Examplel x=st,y=s+t z=exln(y) 6'2 _ 62 5x 62 5y __+__ 5— 5x 5s (33} 5S =exlnyt++e— y ...
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