Use Logarithmic Differentiation to Find the Derivative of Y $$y = \sqrt [ 3 ] { \frac { ( 3 x + 1 ) ( x + 4 ) ^ { 2 } } { \left( x ^ { 4 } + 6 \right) ( x + 8 ) } }$$

question 534

Multiple Choice

Use logarithmic differentiation to find the derivative of y.
- $$y = \sqrt [ 3 ] { \frac { ( 3 x + 1 ) ( x + 4 ) ^ { 2 } } { \left( x ^ { 4 } + 6 \right) ( x + 8 ) } }$$

• A) \(3 \sqrt [ 3 ] { \frac { ( 3 x + 1 ) ( x + 4 ) ^ { 2 } } { \left( x ^ { 4 } + 6 \right) ( x + 8 ) } } \left( \frac { 3 } { 3 x + 1 } + \frac { 2 } { x + 4 } - \frac { 4 x ^ { 3 } } { x ^ { 4 } + 6 } - \frac { 1 } { x + 8 } \right)\)
• B) \(\frac { 1 } { 3 } \left( \ln ( 3 x + 1 ) + 2 \ln ( x + 4 ) - \ln \left( x ^ { 4 } + 6 \right) - \ln ( x + 8 ) \right)\)
• C) \(\frac { 3 } { 3 x + 1 } + \frac { 2 } { x + 4 } - \frac { 4 x ^ { 3 } } { x ^ { 4 } + 6 } - \frac { 1 } { x + 8 }\)
• D) \(\frac { 1 } { 3 } \sqrt [ 3 ] { \frac { ( 3 x + 1 ) ( x + 4 ) ^ { 2 } } { \left( x ^ { 4 } + 6 \right) ( x + 8 ) } } \left( \frac { 3 } { 3 x + 1 } + \frac { 2 } { x + 4 } - \frac { 4 x ^ { 3 } } { x ^ { 4 } + 6 } - \frac { 1 } { x + 8 } \right)\)

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Definitions:

NADPH

A coenzyme involved in redox reactions, providing the reducing power necessary for the biosynthesis of molecules and oxidative stress protection.

ATP

Adenosine triphosphate, a molecule that carries energy within cells, crucial for cellular functions.

Calvin Cycle

A series of biochemical reactions that take place in the stroma of chloroplasts in photosynthetic organisms; it is part of the process of converting carbon dioxide and water into glucose.

Carbon Dioxide Fixation

The process of converting gaseous carbon dioxide into an organic or inorganic form, used by plants and some microorganisms in photosynthesis.