Calculate the derivative of the function. $s ( x ) = \left( \frac { 8 x + 5 } { 2 x - 6 } \right) ^ { 6 }$
A) $s ^ { \prime } ( x ) = - 6 \left( \frac { 8 x + 5 } { 2 x - 6 } \right) ^ { 5 } \frac { 58 x } { ( 2 x - 6 ) ^ { 2 } }$
B) $s ^ { \prime } ( x ) = - 6 \left( \frac { 8 x + 5 } { 2 x - 6 } \right) ^ { 5 } \frac { 48 } { ( 2 x - 6 ) ^ { 2 } }$
C) $s ^ { \prime } ( x ) = \left( \frac { 8 x + 5 } { 2 x - 6 } \right) ^ { 5 } \frac { 58 } { ( 2 x - 6 ) ^ { 2 } }$
D) $s ^ { \prime } ( x ) = - 6 \left( \frac { 8 x + 5 } { 2 x - 6 } \right) ^ { 5 } \frac { 58 } { ( 2 x - 6 ) ^ { 2 } }$
E) $s ^ { \prime } ( x ) = 6 \left( \frac { 8 x + 5 } { 2 x - 6 } \right) ^ { 5 }$

Question

Calculate the derivative of the function.   $s ( x ) = \left( \frac { 8 x + 5 } { 2 x - 6 } \right) ^ { 6 }$  
A)  $s ^ { \prime } ( x ) = - 6 \left( \frac { 8 x + 5 } { 2 x - 6 } \right) ^ { 5 } \frac { 58 x } { ( 2 x - 6 ) ^ { 2 } }$
B) $s ^ { \prime } ( x ) = - 6 \left( \frac { 8 x + 5 } { 2 x - 6 } \right) ^ { 5 } \frac { 48 } { ( 2 x - 6 ) ^ { 2 } }$
C)   $s ^ { \prime } ( x ) = \left( \frac { 8 x + 5 } { 2 x - 6 } \right) ^ { 5 } \frac { 58 } { ( 2 x - 6 ) ^ { 2 } }$
D)   $s ^ { \prime } ( x ) = - 6 \left( \frac { 8 x + 5 } { 2 x - 6 } \right) ^ { 5 } \frac { 58 } { ( 2 x - 6 ) ^ { 2 } }$
E)   $s ^ { \prime } ( x ) = 6 \left( \frac { 8 x + 5 } { 2 x - 6 } \right) ^ { 5 }$

Quizplus · Accepted Answer

To find the derivative of $s(x) = \left( \frac{8x + 5}{2x - 6} \right)^6$, we use the chain rule and the quotient rule. First, differentiate the outer function, which gives $6\left( \frac{8x + 5}{2x - 6} \right)^5$. Then, differentiate the inner function $\frac{8x + 5}{2x - 6}$ using the quotient rule: $\frac{(8)(2x - 6) - (2)(8x + 5)}{(2x - 6)^2}$, which simplifies to $\frac{16x - 48 - 16x - 10}{(2x - 6)^2} = \frac{-58}{(2x - 6)^2}$. Combining these gives the derivative as $-6\left( \frac{8x + 5}{2x - 6} \right)^5 \frac{58}{(2x - 6)^2}$.

Calculate the Derivative of the Function s(x)=(8x+52x−6)6s ( x ) = \left( \frac { 8 x + 5 } { 2 x - 6 } \right) ^ { 6 }s(x)=(2x−68x+5​)6

Calculate the Derivative of the Function $s ( x ) = \left( \frac { 8 x + 5 } { 2 x - 6 } \right) ^ { 6 }$