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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