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

question 137

Multiple Choice

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

• A) \(\frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { x } ^ { 2 } } = \mathrm { ye } ^ { - \mathrm { y } ^ { 2 } } ; \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { y } ^ { 2 } } = 2 x \mathrm { ye } ^ { - \mathrm { y } ^ { 2 } } \left( 2 \mathrm { y } ^ { 2 } - 6 \right) ; \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { y } \partial \mathrm { x } } = \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { x } \partial \mathrm { y } } = \mathrm { e } ^ { - \mathrm { y } ^ { 2 } } \left( 1 - 2 \mathrm { y } ^ { 2 } \right)\)
• B) \(\frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { x } ^ { 2 } } = 0 ; \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { y } ^ { 2 } } = 2 x \mathrm { e } ^ { - \mathrm { y } ^ { 2 } } \left( 2 \mathrm { y } ^ { 2 } - 3 \right) ; \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { y } \partial \mathrm { x } } = \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { x } \partial \mathrm { y } } = \mathrm { e } ^ { - \mathrm { y } ^ { 2 } } \left( 1 - \mathrm { y } ^ { 2 } \right)\)
• C) \(\frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { x } ^ { 2 } } = \mathrm { ye } ^ { - \mathrm { y } ^ { 2 } } ; \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { y } ^ { 2 } } = 2 x y \mathrm { e } ^ { - \mathrm { y } ^ { 2 } } \left( \mathrm { y } ^ { 2 } - 1 \right) ; \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { y } \partial \mathrm { x } } = \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { x } \partial \mathrm { y } } = \mathrm { e } ^ { - \mathrm { y } ^ { 2 } } \left( 1 - \mathrm { y } ^ { 2 } \right)\)
• D) \(\frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { x } ^ { 2 } } = 0 ; \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { y } ^ { 2 } } = 2 \mathrm { xye } ^ { - \mathrm { y } ^ { 2 } } \left( 2 \mathrm { y } ^ { 2 } - 3 \right) ; \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { y } \partial \mathrm { x } } = \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { x } \partial \mathrm { y } } = \mathrm { e } ^ { - \mathrm { y } ^ { 2 } } \left( 1 - 2 \mathrm { y } ^ { 2 } \right)\)

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