exercises6 - z v = e v ln u (c) z u = 2 ue ( u 2-v 2 ) (1 +...

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The chain rule (general version) Find ∂z/∂u and ∂z/∂v by using the chain rule. (a) z = xe - y + ye - x , x = u sin v , y = v cos u (b) z = xe y , x = ln u , y = v (c) z = xe y , x = u 2 + v 2 , y = u 2 - v 2 (d) z = sin( x/y ), x = ln u , y = v Answers: (a) ∂z ∂u = ( e - v cos u - v cos( u ) e - u sin v ) sin v - ( - u sin( v ) e - v cos u + e - u sin v ) v sin u ∂z ∂v = ( e - v cos u - v cos( u ) e - u sin v ) u cos v +( - u sin( v ) e - v cos u + e - u sin v ) cos u (b) ∂z ∂u = e v u
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Unformatted text preview: z v = e v ln u (c) z u = 2 ue ( u 2-v 2 ) (1 + u 2 + v 2 ) z v = 2 ve ( u 2-v 2 ) (1-u 2-v 2 ) (d) z u = 1 vu cos ln u v z v =-ln u v 2 cos ln u v...
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