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# ps1a - Problem Set#1 Answer Key This problem set is...

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2 Partial Derivatives For the following functions, find the partial derivatives with respect to both x and y . 1. f ( x, y ) = xy + y 2 Answer: f x = y, f y = x + 2 y . 2. f ( x, y ) = x 1 3 y 2 3 Answer: f x = 1 3 x 2 3 y 2 3 , f y = 2 3 x 1 3 y 1 3 3. f ( x, y ) = x a y b for constants a and b . Answer: f x = ax a 1 y b , f y = bx a y b 1 . 4. For the function f ( x, y ) = x a y b , develop an expression for the ratio of the two partial derivatives, f x f y , and simplify the result as much as possible. Answer: f x f y = ay bx 5. For the function f ( x, y ) = x a y b , take the natural log of the function and form the partial derivatives. Now, what happens when you form an expression for the ratio of the partial derivatives f x f y ? Answer: This will give you the exact same answer as the previous question. 6. Consider the function f ( x, y ) = ( βx α + δy α ) 1 α , where α, β, and δ are all given constants. Find the partial derivatives of this expression with respect to x and y . Form an expression for the ratio of the two partial derivatives and simplify as much as possible.
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