Find All the Second Order Partial Derivatives of the Given $$f ( x , y ) = \frac { x } { x + y }$$

question 61

Multiple Choice

Find all the second order partial derivatives of the given function.
- $$f ( x , y ) = \frac { x } { x + y }$$

• A) \(\frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { x } ^ { 2 } } = - \frac { 2 \mathrm { y } } { ( \mathrm { x } + \mathrm { y } ) ^ { 3 } } ; \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { y } ^ { 2 } } = \frac { 2 \mathrm { x } } { ( \mathrm { x } + \mathrm { y } ) ^ { 3 } } ; \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { y } \partial \mathrm { x } } = \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { x } \partial \mathrm { y } } = \frac { \mathrm { y } - \mathrm { x } } { ( \mathrm { x } + \mathrm { y } ) ^ { 3 } }\)
• B) \(\frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { x } ^ { 2 } } = - \frac { \mathrm { y } } { ( \mathrm { x } + \mathrm { y } ) ^ { 3 } } ; \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { y } ^ { 2 } } = \frac { \mathrm { x } } { ( \mathrm { x } + \mathrm { y } ) ^ { 3 } } ; \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { y } \partial \mathrm { x } } = \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { x } \partial \mathrm { y } } = \frac { \mathrm { x } - \mathrm { y } } { ( \mathrm { x } + \mathrm { y } ) ^ { 3 } }\)
• C) \(\frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { x } ^ { 2 } } = \frac { 2 \mathrm { y } } { ( \mathrm { x } + \mathrm { y } ) ^ { 3 } ( \mathrm { x } + \mathrm { y } ) ^ { 3 } } ; \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { y } ^ { 2 } } = - \frac { 2 \mathrm { x } } { ( \mathrm { x } + \mathrm { y } ) ^ { 3 } } ; \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { y } \partial \mathrm { x } } = \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { x } \partial \mathrm { y } } = \frac { \mathrm { x } - \mathrm { y } } { ( \mathrm { x } + \mathrm { y } ) ^ { 3 } }\)
• D) \(\frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { x } ^ { 2 } } = - \frac { 2 \mathrm { y } } { ( \mathrm { x } + \mathrm { y } ) ^ { 3 } } ; \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { y } ^ { 2 } } = \frac { 2 \mathrm { x } } { ( \mathrm { x } + \mathrm { y } ) ^ { 3 } } ; \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { y } \partial \mathrm { x } } = \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { x } \partial \mathrm { y } } = \frac { \mathrm { x } - \mathrm { y } } { ( \mathrm { x } + \mathrm { y } ) ^ { 3 } }\)

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Fixed Costs

Costs that do not vary with the quantity of output produced

Monopolistically Competitive

A market structure characterized by many firms offering products that are similar but not perfect substitutes, leading to competitive yet differentiated markets.

Barriers to Entry

Economic, procedural, or regulatory obstacles that prevent new competitors from easily entering an industry or area of business.

Monopolistically Competitive

In a monopolistically competitive market, many firms sell products that are similar but not identical, allowing for some degree of market power and price setting.