Find the derivative of y with respect to the independent variable.
-$$\left. y = \log 7 \left( \frac { x + 6 } { x - 6 } \right) ^ { \ln 7 } \right)$$
A) $\frac { - 12 } { ( x + 6 ) ( x - 6 ) }$
B) $\frac { 1 } { ( x - 6 ) ^ { 2 } }$
C) $\frac { - 2 } { x + 6 }$
D) $\left( \frac { 1 } { \ln 7 } \left( \frac { x + 6 } { x - 6 } \right) ^ { \ln 7 } \right.$

Question

Quizplus · Accepted Answer

The derivative of $y = \log 7 \left( \frac { x + 6 } { x - 6 } \right) ^ { \ln 7 }$ can be found using the chain rule and the fact that the derivative of $\log_a(x)$ is $\frac{1}{x \ln(a)}$. The expression inside the logarithm is raised to the power of $\ln(7)$, which simplifies the process since $\log_7(x^{\ln 7}) = \ln(x)$ due to the change of base formula. Differentiating $\ln\left(\frac{x+6}{x-6}\right)$ with respect to $x$ gives $\frac{1}{\frac{x+6}{x-6}} \cdot \left(\frac{(x-6)\cdot 1 - (x+6)\cdot 1}{(x-6)^2}\right)$, which simplifies to $\frac{-12}{(x+6)(x-6)}$.

Find the Derivative of Y with Respect to the Independent y=log⁡7(x+6x−6)ln⁡7)\left. y = \log 7 \left( \frac { x + 6 } { x - 6 } \right) ^ { \ln 7 } \right)y=log7(x−6x+6​)ln7)

Find the Derivative of Y with Respect to the Independent $\left. y = \log 7 \left( \frac { x + 6 } { x - 6 } \right) ^ { \ln 7 } \right)$