$\text { Find } f ^ {t } ( x ) \text { if } f ( x ) = \log _ { 5 } \left( \frac { x ^ { 3 } - 1 } { x ^ { 3 } - 6 } \right)$
A)$f ^ { t } ( x ) = \frac { x ^ { 2 } - 5 } { \left( x ^ { 3 } - 1 \right) \left( x ^ { 2 } - 6 \right) }$
B) $f ^ { t } ( x ) = \frac { \left( x ^ { 3 } - 1 \right) } { \left( x ^ { 3 } - 6 \right) \ln ( 5 ) }$
C)$f ^ { t } ( x ) = \frac { - 15 x ^ { 2 } } { \left( x ^ { 3 } - 1 \right) \left( x ^ { 3 } - 6 \right) }$
D)$f ^ { t } ( x ) = \frac { x ^ { 3 } - 6 } { \left( x ^ { 3 } - 1 \right) \ln ( 5 ) }$
E)$f ^ { t } ( x ) = \frac { - 15 x ^ { 2 } } { \left( x ^ { 3 } - 1 \right) \left( x ^ { 3 } - 6 \right) \ln ( 5 ) }$

Question

$\text { Find } f ^ {t } ( x ) \text { if } f ( x ) = \log _ { 5 } \left( \frac { x ^ { 3 } - 1 } { x ^ { 3 } - 6 } \right)$ 
A)$f ^ { t } ( x ) = \frac { x ^ { 2 } - 5 } { \left( x ^ { 3 } - 1 \right) \left( x ^ { 2 } - 6 \right) }$
B) $f ^ { t } ( x ) = \frac { \left( x ^ { 3 } - 1 \right) } { \left( x ^ { 3 } - 6 \right) \ln ( 5 ) }$
C)$f ^ { t } ( x ) = \frac { - 15 x ^ { 2 } } { \left( x ^ { 3 } - 1 \right) \left( x ^ { 3 } - 6 \right) }$
D)$f ^ { t } ( x ) = \frac { x ^ { 3 } - 6 } { \left( x ^ { 3 } - 1 \right) \ln ( 5 ) }$
E)$f ^ { t } ( x ) = \frac { - 15 x ^ { 2 } } { \left( x ^ { 3 } - 1 \right) \left( x ^ { 3 } - 6 \right) \ln ( 5 ) }$

Quizplus · Accepted Answer

The derivative of $f(x) = \log_5\left(\frac{x^3 - 1}{x^3 - 6}\right)$ involves applying the chain rule and the derivative of a logarithm. The correct derivative, considering the base of the logarithm and the quotient inside it, is $\frac{-15x^2}{(x^3 - 1)(x^3 - 6)\ln(5)}$, which matches option E.

Find ft(x) if f(x)=log⁡5(x3−1x3−6)\text { Find } f ^ {t } ( x ) \text { if } f ( x ) = \log _ { 5 } \left( \frac { x ^ { 3 } - 1 } { x ^ { 3 } - 6 } \right) Find ft(x) if f(x)=log5​(x3−6x3−1​)

$\text { Find } f ^ {t } ( x ) \text { if } f ( x ) = \log _ { 5 } \left( \frac { x ^ { 3 } - 1 } { x ^ { 3 } - 6 } \right)$