and y y z El z z y x x z El z z x 118 Partial elasticities x x El f x f x f x

And y y z el z z y x x z el z z x 118 partial

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and y y z El z z y = x x z El z z x = 11.8 Partial elasticities ( ) '( ) ( ) x x El f x f x f x = If f is differentiable in x and f ( x ) 0 1. Function of one variable: 2. Function of 2 variables: x x z El z z x = = = = ( ) , 0 a f x Ax x = > = > = = 1 ( ) a x a x El f x Aax Ax - = a = ( ) ; , 0 a b f x Ax y x y = > 1 b y a a b x Aax Ax y - = a = 11.2 Higher-order partial derivatives Find the second-order partial derivatives of the following function: 2 2 ( ) f f x dx dx = 2 2 (2 1) 2 xy y x = + = 2 2 3 ( , ) f x y x y x y = + + = + 2 ( ) f f y x y dx = 2 (2 1) xy y = + 4 xy = 2 ( ) f f x y dx dy = ∂ ∂ ∂ ∂ 2 2 (2 3 ) x y y x = + 4 xy = 2 2 ( ) f f y dy dy = 2 2 (2 3 ) x y y y = + + 2 2 6 x y = + = + = = 11.1 Higher-order partial derivatives You can also use the following notation: For most functions these two ‘mixed’ second- order partial derivatives (or ‘cross-partials) are equal. 12 '' ( , ) f x y 21 '' ( , ) f x y 11.1 Example 5b/page 380 Determine the domain of the function given by the following formula and draw the set in the xy -plane. 2 2 2 2 2 ( , ) 9 ( ) 4 g x y x y x y = + - + = + - + + + + - + - 2 2 Domain: 4 9 x y < + 2 2 2 is the graph consisting all the points on the circle with centre at the origin and radius . x y r r + = +
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4 11.1 Example 2 2 2 2 2 ( , ) 9 ( ) 4 g x y x y x y = + - + + + - - 2 2 Domain: 4 9 x y < + 6.11 Example Find the domain and compute the derivative: y is defined for
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