Use logarithmic differentiation to find $f ^ { \prime } ( x )$ . $$f ( x ) = \sqrt [ 3 ] { \frac { 6 x + 3 } { 4 + 4 x } }$$
A) $f ^ { \prime } ( x ) = \left( \frac { 6 } { 6 x + 3 } - \frac { 4 } { 4 + 4 x } \right) f ( x )$
B) $f ^ { \prime } ( x ) = \frac { 1 } { 3 } \left( \frac { 6 } { 6 x + 3 } - \frac { 4 } { 4 + 4 x } \right) f ( x )$
C) $f ^ { \prime } ( x ) = \left( \frac { 6 x + 3 } { 4 + 4 x } \right) ^ { - 2 / 3 }$
D) $f ^ { \prime } ( x ) = f ( x ) \left( \frac { 6 x + 3 } { 4 + 4 x } \right) ^ { - 2 / 3 }$

Question

Quizplus · Accepted Answer

Logarithmic differentiation simplifies the differentiation of functions in the form of one function raised to another. For $f(x) = \sqrt[3]{\frac{6x+3}{4+4x}}$, taking the natural logarithm on both sides and differentiating with respect to $x$ leads to the derivative $f'(x) = \frac{1}{3}\left(\frac{6}{6x+3} - \frac{4}{4+4x}\right)f(x)$, where the factor $\frac{1}{3}$ comes from the power rule applied during differentiation.

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

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