Use logarithmic differentiation to find the derivative of y.
-$$y = \frac { x \sqrt { x ^ { 5 } + 1 } } { ( x + 6 ) ^ { 2 / 3 } }$$
A) $\ln x + \frac { 1 } { 2 } \ln \left( x ^ { 5 } + 1 \right) - \frac { 2 } { 3 } \ln ( x + 6 )$
B) $\frac { x \sqrt { x ^ { 5 } + 1 } } { ( x + 6 ) ^ { 2 / 3 } } \left( \ln x + \frac { 1 } { 2 } \ln \left( x ^ { 5 } + 1 \right) - \frac { 2 } { 3 } \ln ( x + 6 ) \right)$
C) $\frac { 1 } { x } + \frac { 5 x ^ { 4 } } { 2 x ^ { 5 } + 2 } - \frac { 2 } { 3 x + 18 }$
D) $\frac { x \sqrt { x ^ { 5 } + 1 } } { ( x + 6 ) ^ { 2 / 3 } } \left( \frac { 1 } { x } + \frac { 5 x ^ { 4 } } { 2 x ^ { 5 } + 2 } - \frac { 2 } { 3 x + 18 } \right)$

Question

Quizplus · Accepted Answer

Logarithmic differentiation simplifies the differentiation process for complex functions by taking the natural logarithm of both sides, differentiating, and then solving for the derivative. The correct answer involves applying the product rule and chain rule within the logarithmic differentiation framework, leading to the expression in choice D, which correctly represents the derivative of the given function.

Use Logarithmic Differentiation to Find the Derivative of Y y=xx5+1(x+6)2/3y = \frac { x \sqrt { x ^ { 5 } + 1 } } { ( x + 6 ) ^ { 2 / 3 } }y=(x+6)2/3xx5+1​​

Use Logarithmic Differentiation to Find the Derivative of Y $y = \frac { x \sqrt { x ^ { 5 } + 1 } } { ( x + 6 ) ^ { 2 / 3 } }$