Find All the Second Order Partial Derivatives of the Given $$f ( x , y ) = \ln \left( x ^ { 2 } y - x \right)$$

question 317

Multiple Choice

Find all the second order partial derivatives of the given function.
- $$f ( x , y ) = \ln \left( x ^ { 2 } y - x \right)$$

• A) \(\frac { \partial ^ { 2 } f } { \partial x ^ { 2 } } = \frac { 2 x y - 2 x ^ { 2 } y ^ { 2 } - 1 } { \left( x ^ { 2 } y - x \right) ^ { 2 } } ; \frac { \partial ^ { 2 } f } { \partial y ^ { 2 } } = - \frac { x ^ { 4 } } { \left( x ^ { 2 } y - x \right) ^ { 2 } } ; \frac { \partial ^ { 2 } f } { \partial y \partial x } = \frac { \partial ^ { 2 } f } { \partial x \partial y } = - \frac { x ^ { 2 } } { \left( x ^ { 2 } y - x \right) ^ { 2 } }\)
• B) \(\frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { x } ^ { 2 } } = \frac { 2 \mathrm { xy } - 2 \mathrm { x } ^ { 2 } \mathrm { y } ^ { 2 } - 1 } { \left( \mathrm { x } ^ { 2 } \mathrm { y } - \mathrm { x } \right) ^ { 2 } } ; \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { y } ^ { 2 } } = - \frac { \mathrm { x } ^ { 2 } } { \left( \mathrm { x } ^ { 2 } \mathrm { y } - \mathrm { x } \right) ^ { 2 } } ; \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { y } \partial \mathrm { x } } = \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { x } \partial \mathrm { y } } = - \frac { \mathrm { x } ^ { 2 } } { \left( \mathrm { x } ^ { 2 } \mathrm { y } - \mathrm { x } \right) ^ { 2 } }\)
• C) \(\frac { \partial ^ { 2 } f } { \partial x ^ { 2 } } = \frac { x y - x ^ { 2 } y ^ { 2 } - 1 } { \left( x ^ { 2 } y - x \right) ^ { 2 } } ; \frac { \partial ^ { 2 } f } { \partial y ^ { 2 } } = - \frac { x ^ { 4 } } { \left( x ^ { 2 } y - x \right) ^ { 2 } } ; \frac { \partial ^ { 2 } f } { \partial y \partial x } = \frac { \partial ^ { 2 } f } { \partial x \partial y } = - \frac { x ^ { 2 } } { \left( x ^ { 2 } y - x \right) ^ { 2 } }\)
• D) \(\frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { x } ^ { 2 } } = \frac { 2 \mathrm { xy } - 2 \mathrm { x } ^ { 2 } \mathrm { y } ^ { 2 } - 1 } { \left( \mathrm { x } ^ { 2 } \mathrm { y } - \mathrm { x } \right) ^ { 2 } } ; \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { y } ^ { 2 } } = \frac { \mathrm { x } ^ { 4 } } { \left( \mathrm { x } ^ { 2 } \mathrm { y } - \mathrm { x } \right) ^ { 2 } } ; \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { y } \partial \mathrm { x } } = \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { x } \partial \mathrm { y } } = \frac { \mathrm { x } ^ { 2 } } { \left( \mathrm { x } ^ { 2 } \mathrm { y } - \mathrm { x } \right) ^ { 2 } }\)

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