Find the Derivative of the Function $g ( x ) = \left( \frac { x + 6 } { x ^ { 2 } + 7 } \right) ^ { 4 }$

question 123

Multiple Choice

Find the derivative of the function. $g ( x ) = \left( \frac { x + 6 } { x ^ { 2 } + 7 } \right) ^ { 4 }$

• A) \(g ^ { t } ( x ) = \frac { 4 \left( 7 - 12 x + x ^ { 2 } \right) } { ( 6 + x ) \left( 7 + x ^ { 2 } \right) } \left( \frac { 6 + x } { 7 + x ^ { 2 } } \right) ^ { 4 }\)
• B)\(g ^ { t } ( x ) = \frac { 4 \left( 7 - 12 x - x ^ { 2 } \right) ( 6 + x ) ^ { 3 } } { \left( 7 + x ^ { 2 } \right) ^ { 5 } }\)
• C)\(g ^ { t } ( x ) = \frac { 4 \left( 7 + 12 x - x ^ { 2 } \right) ( 6 + x ) ^ { 3 } } { \left( 7 + x ^ { 2 } \right) ^ { 5 } }\)
• D)\(g ^ { t } ( x ) = \frac { 4 \left( 7 - 12 x - x ^ { 2 } \right) ( 6 + x ) ^ { 5 } } { \left( 7 + x ^ { 2 } \right) ^ { 3 } }\)
• E)\(g ^ { t } ( x ) = \frac { 4 \left( 7 - 12 x - x ^ { 2 } \right) ( 6 + x ) ^ { 3 } } { \left( 7 + x ^ { 2 } \right) ^ { 5 } }\).

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Definitions:

Multiplier Size

The ratio of change in the economic output to the initial change in spending or investment that brought it about, indicating the scale of impact on the economy.

Government Purchases

Expenditures on goods and services by government at all levels, excluding transfer payments.

Total Output

The total value of all goods and services produced in an economy during a specific time period.

Government Purchases

Involves spending by the government on goods and services that directly contribute to the economy's aggregate demand.