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# week06 - Math 23 Sections 110-113 B Dodson Week 6 Homework...

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Math 23 Sections 110-113 B. Dodson Week 6 Homework: 14.3 partial derivatives, 2nd order deriv 14.4 tangent plane, diﬀerentials 14.5 chain rule Problem 14.3.15: Find the partial derivatives of the function z = f ( x, y ) = xe 3 y . Find f x (2 , 1) . Solution: For f x we (temporarily) hold y constant, so e 3 y constant, giving f as constant · x . We then take the derivative in x , with (constant · x )’ = constant, or f x = e 3 y . Likewise, with x constant, f y = x ∂y ( e 3 y ) = x d dy ( e 3 y ) = 3 xe 3 y . Finally, f x (2 , 1) = e 3 gives the rate of change of f at (2 , 1) with respect to x ; which we may constrast with f y (2 , 1) = 6 e 3 , the rate of change of f at (2 , 1) with respect to y .

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2 . For example, f is growing six times more rapidly in y than in x . We also solved #49, 14.3, and in particular veriﬁed that the second partials z xy = z yx . Week 6 Homework: 14.4 tangent plane, diﬀerentials 14.5 chain rule Problem 14.4.3: Find the tangent plane to the surface
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week06 - Math 23 Sections 110-113 B Dodson Week 6 Homework...

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