Find All the Second Order Partial Derivatives of the Given $f ( x , y ) = x \ln ( y - x )$

question 262

Multiple Choice

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

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

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