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math192_hw4sp06

# math192_hw4sp06 - HW4 Solutions 14.3 Partial Derivatives 4...

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HW4 Solutions 14.3 Partial Derivatives 4. ∂f ∂x = 5 y - 14 x + 3, ∂f ∂y = 5 x - 2 y - 6 14. ∂f ∂x = - e - x sin( x + y ) + e - x cos( x + y ), ∂f ∂y = e - x cos( x + y ) 18. ∂f ∂x = - 6 cos(3 x - y 2 ) sin(3 x - y 2 ), ∂f ∂y = 4 y cos(3 x - y 2 ) sin(3 x - y 2 ) 30. f x = yz x , f y = z ln( xy ) + z , f z = y ln( xy ) 46. ∂s ∂x = - y x 2 + y 2 , ∂s ∂y = x x 2 + y 2 , 2 s ∂x 2 = 2 xy ( x 2 + y 2 ) 2 , 2 s ∂y 2 = - 2 xy ( x 2 + y 2 ) 2 2 s ∂y∂x = 2 s ∂x∂y = y 2 - x 2 ( x 2 + y 2 ) 2 66. ∂f ∂x = x x 2 + y 2 , ∂f ∂y = y x 2 + y 2 2 f ∂x 2 = y 2 - x 2 ( x 2 + y 2 ) 2 , 2 f ∂y 2 = x 2 - y 2 ( x 2 + y 2 ) 2 , 2 f ∂x 2 + 2 f ∂y 2 = y 2 - x 2 ( x 2 + y 2 ) 2 + x 2 - y 2 ( x 2 + y 2 ) 2 = 0

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14.4 The Chain Rule 2. (a) ∂w ∂x = 2 x , ∂w ∂y = 2 y , ∂x ∂t = - sin t + cos t , dy dt = - sin t - cos t dw dt = (2 x )( - sin t + cos t ) + (2 y )( - sin t - cos t ) = 0; w = x 2 + y 2 = (cos t +sin t ) 2 +(cos t - sin t ) 2 = 2 cos 2 t +2 sin 2 t = 2 dw dt = 0 (b) dw dt (0) = 0 6. (a) ∂w ∂x = - y cos xy , ∂w ∂y = - x cos xy , ∂w ∂z = 1, ∂x ∂t = 1, dy dt = dy dt = 1 t , dz dt = e t - 1 dw dt = - (ln t )[cos( t ln t )] - cos( t ln t ) + e t - 1 w = z - sin xy = e t - 1 - sin( t ln t ) dw dt = e t - 1 - (1 + ln t ) cos( t ln t ) (b) dw dt (1) = 0 10. (a) ∂w ∂u =
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