Use logarithmic differentiation to find $f ^ { \prime } ( x )$ . $f ( x ) = \sqrt [ 8 ] { \frac { 6 x + 5 } { 8 + 6 x } }$
A) $f ^ { \prime } ( x ) = \frac { 1 } { 8 } \left( \frac { 6 } { 6 x + 5 } - \frac { 6 } { 8 + 6 x } \right) f ( x )$
B) $f ^ { \prime } ( x ) = f ( x ) \left( \frac { 6 x + 5 } { 8 + 6 x } \right) ^ { - 7 / 8 }$
C) $f ^ { \prime } ( x ) = \left( \frac { 6 } { 6 x + 5 } - \frac { 6 } { 8 + 6 x } \right) f ( x )$
D) $f ^ { \prime } ( x ) = \left( \frac { 6 x + 5 } { 8 + 6 x } \right) ^ { - 7 / 8 }$

Question

Quizplus · Accepted Answer

Logarithmic differentiation simplifies the differentiation of functions that are products, quotients, or powers by taking the natural logarithm of both sides, differentiating implicitly, and then solving for the derivative. For $f(x) = \sqrt[8]{\frac{6x + 5}{8 + 6x}}$, applying logarithmic differentiation and simplifying leads to the derivative in option A.

Use Logarithmic Differentiation to Find f′(x)f ^ { \prime } ( x )f′(x)

Use Logarithmic Differentiation to Find $f ^ { \prime } ( x )$