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) $\frac { 1 } { 3 } \sqrt [ 3 ] { \frac { ( 3 x + 1 ) ( x + 4 ) ^ { 2 } } { \left( x ^ { 4 } + 6 \right) ( x + 8 ) } } \left( \frac { 3 } { 3 x + 1 } + \frac { 2 } { x + 4 } - \frac { 4 x ^ { 3 } } { x ^ { 4 } + 6 } - \frac { 1 } { x + 8 } \right)$

Question

Quizplus · Accepted Answer

Logarithmic differentiation involves taking the natural log of both sides, simplifying using log properties, differentiating, and then solving for dy/dx. The correct derivative is option D.

Use Logarithmic Differentiation to Find the Derivative of Y y=(3x+1)(x+4)2(x4+6)(x+8)3y = \sqrt [ 3 ] { \frac { ( 3 x + 1 ) ( x + 4 ) ^ { 2 } } { \left( x ^ { 4 } + 6 \right) ( x + 8 ) } }y=3(x4+6)(x+8)(3x+1)(x+4)2​​

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