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) \(\frac { \partial ^ { 2 } f } { \partial x ^ { 2 } } = \frac { e ^ { x / y } } { y ^ { 2 } } ; \frac { \partial ^ { 2 } f } { \partial y ^ { 2 } } = e ^ { x / y } \left( \frac { x ^ { 2 } + 2 x y } { y ^ { 4 } } \right) ; \frac { \partial ^ { 2 } f } { \partial y \partial x } = \frac { \partial ^ { 2 } f } { \partial x \partial y } = - e ^ { x / y } \left( \frac { y + x } { y ^ { 3 } } \right)\)

Distinguish between fiat currencies and cryptocurrencies.

Understand the role and mechanism of currency exchange in international trade.

Grasp the concept of floating and managed exchange rates.

Recognize the effects of economic policies and market movements on currency value.

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