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

question 317

Multiple Choice

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

• A) \(\frac { \partial ^ { 2 } f } { \partial x ^ { 2 } } = \frac { 2 x y - 2 x ^ { 2 } y ^ { 2 } - 1 } { \left( x ^ { 2 } y - x \right) ^ { 2 } } ; \frac { \partial ^ { 2 } f } { \partial y ^ { 2 } } = - \frac { x ^ { 4 } } { \left( x ^ { 2 } y - x \right) ^ { 2 } } ; \frac { \partial ^ { 2 } f } { \partial y \partial x } = \frac { \partial ^ { 2 } f } { \partial x \partial y } = - \frac { x ^ { 2 } } { \left( x ^ { 2 } y - x \right) ^ { 2 } }\)
• B) \(\frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { x } ^ { 2 } } = \frac { 2 \mathrm { xy } - 2 \mathrm { x } ^ { 2 } \mathrm { y } ^ { 2 } - 1 } { \left( \mathrm { x } ^ { 2 } \mathrm { y } - \mathrm { x } \right) ^ { 2 } } ; \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { y } ^ { 2 } } = - \frac { \mathrm { x } ^ { 2 } } { \left( \mathrm { x } ^ { 2 } \mathrm { y } - \mathrm { x } \right) ^ { 2 } } ; \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { y } \partial \mathrm { x } } = \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { x } \partial \mathrm { y } } = - \frac { \mathrm { x } ^ { 2 } } { \left( \mathrm { x } ^ { 2 } \mathrm { y } - \mathrm { x } \right) ^ { 2 } }\)
• C) \(\frac { \partial ^ { 2 } f } { \partial x ^ { 2 } } = \frac { x y - x ^ { 2 } y ^ { 2 } - 1 } { \left( x ^ { 2 } y - x \right) ^ { 2 } } ; \frac { \partial ^ { 2 } f } { \partial y ^ { 2 } } = - \frac { x ^ { 4 } } { \left( x ^ { 2 } y - x \right) ^ { 2 } } ; \frac { \partial ^ { 2 } f } { \partial y \partial x } = \frac { \partial ^ { 2 } f } { \partial x \partial y } = - \frac { x ^ { 2 } } { \left( x ^ { 2 } y - x \right) ^ { 2 } }\)
• D) \(\frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { x } ^ { 2 } } = \frac { 2 \mathrm { xy } - 2 \mathrm { x } ^ { 2 } \mathrm { y } ^ { 2 } - 1 } { \left( \mathrm { x } ^ { 2 } \mathrm { y } - \mathrm { x } \right) ^ { 2 } } ; \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { y } ^ { 2 } } = \frac { \mathrm { x } ^ { 4 } } { \left( \mathrm { x } ^ { 2 } \mathrm { y } - \mathrm { x } \right) ^ { 2 } } ; \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { y } \partial \mathrm { x } } = \frac { \partial ^ { 2 } \mathrm { f } } { \partial \mathrm { x } \partial \mathrm { y } } = \frac { \mathrm { x } ^ { 2 } } { \left( \mathrm { x } ^ { 2 } \mathrm { y } - \mathrm { x } \right) ^ { 2 } }\)

Recognize the impact of genetic disorders and how some can be managed or influenced by lifestyle or diet.

Identify the typical chromosomal patterns associated with different genders and their abnormalities.

Comprehend the biological mechanisms of heredity including the role of chromosomes, genes, and DNA.

Appreciate the genetic basis of physical and personality traits.

---

Definitions:

Cerebellum

A part of the brain located at the back of the skull, crucial for coordination and precision of voluntary movements, as well as balance.

Coordinated Actions

Organized and synchronized activities that are directed towards achieving a specific goal.

Hindbrain

The posterior part of the brain, including the cerebellum, pons, and medulla, involved in regulating basic life functions like heartbeat and breathing.

Reflexes

Involuntary and nearly instantaneous movements in response to stimuli, which are mediated through the nervous system without conscious thought.