Find All the Second Order Partial Derivatives of the Given $f ( x , y ) = e ^ { x / y }$

question 276

Multiple Choice

Find all the second order partial derivatives of the given function.
- $f ( x , y ) = e ^ { x / y }$

• A) \(\frac { \partial ^ { 2 } f } { \partial x ^ { 2 } } = \frac { e ^ { x / y } } { y ^ { 2 } } ; \frac { \partial ^ { 2 } f } { \partial y ^ { 2 } } = e ^ { x / y } \left( \frac { x ^ { 2 } + 2 x y } { y ^ { 3 } } \right) ; \frac { \partial ^ { 2 } f } { \partial y \partial x } = \frac { \partial ^ { 2 } f } { \partial x \partial y } = - e ^ { x / y } \left( \frac { y + x } { y ^ { 3 } } \right)\)
• B) \(\frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { x } ^ { 2 } } = \frac { \mathrm { e } ^ { \mathrm { x } / \mathrm { y } } } { \mathrm { y } ^ { 2 } } ; \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { y } ^ { 2 } } = \left( \frac { \mathrm { x } ^ { 2 } + 2 \mathrm { xy } } { \mathrm { y } ^ { 4 } } \right) ; \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { y } \partial \mathrm { x } } = \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { x } \partial \mathrm { y } } = \frac { \mathrm { y } + \mathrm { x } } { \mathrm { y } ^ { 3 } }\)
• C) \(\frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { x } ^ { 2 } } = \frac { \mathrm { e } ^ { \mathrm { x } / \mathrm { y } } } { \mathrm { y } ^ { 2 } } ; \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { y } ^ { 2 } } = - \mathrm { e } ^ { \mathrm { x } / \mathrm { y } } \left( \frac { \mathrm { x } ^ { 2 } + 2 \mathrm { xy } } { \mathrm { y } ^ { 3 } } \right) ; \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { y } \partial \mathrm { x } } = \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { x } \partial \mathrm { y } } = \mathrm { e } ^ { \mathrm { x } / \mathrm { y } } \left( \frac { \mathrm { y } + \mathrm { x } } { \mathrm { y } ^ { 3 } } \right)\)
• D) \(\frac { \partial ^ { 2 } f } { \partial x ^ { 2 } } = \frac { e ^ { x / y } } { y ^ { 2 } } ; \frac { \partial ^ { 2 } f } { \partial y ^ { 2 } } = e ^ { x / y } \left( \frac { x ^ { 2 } + 2 x y } { y ^ { 4 } } \right) ; \frac { \partial ^ { 2 } f } { \partial y \partial x } = \frac { \partial ^ { 2 } f } { \partial x \partial y } = - e ^ { x / y } \left( \frac { y + x } { y ^ { 3 } } \right)\)

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