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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Definitions:

Somatic Nervous System

Part of the peripheral nervous system that handles voluntary control of body movements via skeletal muscles.

Central Nervous System

The part of the nervous system consisting of the brain and spinal cord, responsible for processing and sending out information to and from all parts of the body.

Sympathetic Nervous System

Part of the autonomic nervous system that prepares the body for fight or flight responses by increasing heart rate, dilating pupils, and other physiological changes.

Endocrine System

A network of glands in the body that secrete hormones directly into the bloodstream to regulate bodily functions.