Find All the Second Order Partial Derivatives of the Given $$\mathrm { f } ( \mathrm { x } , \mathrm { y } ) = x \mathrm { y } ^ { 2 } + y \mathrm { e } ^ { \mathrm { x } ^ { 2 } + 5 }$$

question 74

Multiple Choice

Find all the second order partial derivatives of the given function.
- $$\mathrm { f } ( \mathrm { x } , \mathrm { y } ) = x \mathrm { y } ^ { 2 } + y \mathrm { e } ^ { \mathrm { x } ^ { 2 } + 5 }$$

• A) \(\frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { x } ^ { 2 } } = 2 y \mathrm { e } ^ { \mathrm { x } ^ { 2 } } ; \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { y } ^ { 2 } } = 2 \mathrm { x } ; \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { y } \partial \mathrm { x } } = \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { x } \partial \mathrm { y } } = 2 \mathrm { y } + 2 x e ^ { \mathrm { x } ^ { 2 } }\)
• B) \(\frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { x } ^ { 2 } } = 2 y \mathrm { e } ^ { x ^ { 2 } } ; \frac { \partial ^ { 2 } \mathrm { f } } { \partial y ^ { 2 } } = 2 \mathrm { x } ; \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { y } \partial \mathrm { x } } = \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { x } \partial \mathrm { y } } = 2 x \mathrm { e } ^ { \mathrm { x } ^ { 2 } }\)
• C) \(\frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { x } ^ { 2 } } = y \mathrm { e } ^ { \mathrm { x } ^ { 2 } } \left( 1 + 2 \mathrm { x } ^ { 2 } \right) ; \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { y } ^ { 2 } } = \mathrm { x } ; \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { y } \partial \mathrm { x } } = \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { x } \partial \mathrm { y } } = \mathrm { y } + x \mathrm { x } ^ { 2 }\)
• D) \(\frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { x } ^ { 2 } } = 2 \mathrm { ye } ^ { \mathrm { x } ^ { 2 } } \left( 1 + 2 \mathrm { x } ^ { 2 } \right) ; \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { y } ^ { 2 } } = 2 \mathrm { x } ; \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { y } \partial \mathrm { x } } = \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { x } \partial \mathrm { y } } = 2 \mathrm { y } + 2 \mathrm { xe } ^ { \mathrm { x } ^ { 2 } }\)

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