Solve the Problem $f ( x , y ) = \tan ^ { - 1 } \frac { y } { x }$

question 124

Multiple Choice

Solve the problem.
-Find the derivative of the function $f ( x , y ) = \tan ^ { - 1 } \frac { y } { x }$ at the point $( - 4,4 )$ in the direction in which the function decreases most rapidly.

• A) \(- \frac { \sqrt { 2 } } { 12 }\)
• B) \(- \frac { \sqrt { 3 } } { 8 }\)
• C) \(- \frac { \sqrt { 2 } } { 8 }\)
• D) \(- \frac { \sqrt { 3 } } { 12 }\)

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Definitions:

Easy Entry And Exit

A characteristic of competitive markets where there are minimal barriers for new firms to enter or exit the market.

Perfect Competition

A theoretical market structure that features no barriers to entry, a large number of sellers and buyers, completely homogenous products, and perfect information available to all market participants.

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A situation where industries consistently operate below their full output capacity due to prolonged demand shortfall.

Monopolistic Competition

A market structure characterized by many firms selling products that are similar but not identical, allowing for competition based on price, quality, and marketing.