Find All the Second Order Partial Derivatives of the Given $f ( x , y ) = x \ln ( y - x )$

question 262

Multiple Choice

Find all the second order partial derivatives of the given function.
- $f ( x , y ) = x \ln ( y - x )$

• A) \(\frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { x } ^ { 2 } } = \frac { 2 \mathrm { y } - \mathrm { x } } { ( \mathrm { y } - \mathrm { x } ) ^ { 2 } } ; \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { y } ^ { 2 } } = - \frac { \mathrm { x } } { ( \mathrm { y } - \mathrm { x } ) ^ { 2 } } ; \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { y } \partial \mathrm { x } } = \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { x } \partial \mathrm { y } } = \frac { \mathrm { y } } { ( \mathrm { y } - \mathrm { x } ) ^ { 2 } }\)
• B) \(\frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { x } ^ { 2 } } = \frac { \mathrm { x } - 2 \mathrm { y } } { ( \mathrm { y } - \mathrm { x } ) ^ { 2 } } ; \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { y } ^ { 2 } } = \frac { \mathrm { x } } { ( \mathrm { y } - \mathrm { x } ) ^ { 2 } } ; \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { y } \partial \mathrm { x } } = \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { x } \partial \mathrm { y } } = \frac { \mathrm { y } } { ( \mathrm { y } - \mathrm { x } ) ^ { 2 } }\)
• C) \(\frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { x } ^ { 2 } } = \frac { \mathrm { x } - 2 \mathrm { y } } { ( \mathrm { y } - \mathrm { x } ) ^ { 2 } } ; \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { y } ^ { 2 } } = - \frac { \mathrm { x } } { ( \mathrm { y } - \mathrm { x } ) ^ { 2 } } ; \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { y } \partial \mathrm { x } } = \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { x } \partial \mathrm { y } } = - \frac { \mathrm { y } } { ( \mathrm { y } - \mathrm { x } ) ^ { 2 } }\)
• D) \(\frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { x } ^ { 2 } } = \frac { \mathrm { x } - 2 \mathrm { y } } { ( \mathrm { y } - \mathrm { x } ) ^ { 2 } } ; \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { y } ^ { 2 } } = - \frac { \mathrm { x } } { ( \mathrm { y } - \mathrm { x } ) ^ { 2 } } ; \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { y } \partial \mathrm { x } } = \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { x } \partial \mathrm { y } } = \frac { \mathrm { y } } { ( \mathrm { y } - \mathrm { x } ) ^ { 2 } }\)

Recognize alternative imaging methods based on specific requirements, such as the availability of radioactive tracers.

Understand the principle and purpose of lesioning in studying brain function.

Identify the brain-imaging methods that utilize radioactive substances to map brain activity.

Differentiate between imaging techniques based on their technological foundations (X-rays, radio pulses, magnetic fields).

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Definitions:

Deuterostomes

A superphylum of animals characterized by the blastopore developing into the anus during early embryo development.

Blastopore

The opening of the central cavity of an embryo in the early stage of development, which eventually develops into the digestive tract in some animals.

Coelom

The fluid-filled body cavity between the intestinal canal and the body wall in triploblastic animals, providing space for the development of internal organs.

Protostomes

A group of animals characterized by developing the mouth before the anus during embryonic development.