Find the derivative of the function. $g ( x ) = \left( \frac { x + 6 } { x ^ { 2 } + 7 } \right) ^ { 4 }$
A) $g ^ { t } ( x ) = \frac { 4 \left( 7 - 12 x + x ^ { 2 } \right) } { ( 6 + x ) \left( 7 + x ^ { 2 } \right) } \left( \frac { 6 + x } { 7 + x ^ { 2 } } \right) ^ { 4 }$
B)$g ^ { t } ( x ) = \frac { 4 \left( 7 - 12 x - x ^ { 2 } \right) ( 6 + x ) ^ { 3 } } { \left( 7 + x ^ { 2 } \right) ^ { 5 } }$
C)$g ^ { t } ( x ) = \frac { 4 \left( 7 + 12 x - x ^ { 2 } \right) ( 6 + x ) ^ { 3 } } { \left( 7 + x ^ { 2 } \right) ^ { 5 } }$
D)$g ^ { t } ( x ) = \frac { 4 \left( 7 - 12 x - x ^ { 2 } \right) ( 6 + x ) ^ { 5 } } { \left( 7 + x ^ { 2 } \right) ^ { 3 } }$
E)$g ^ { t } ( x ) = \frac { 4 \left( 7 - 12 x - x ^ { 2 } \right) ( 6 + x ) ^ { 3 } } { \left( 7 + x ^ { 2 } \right) ^ { 5 } }$.

Question

Quizplus · Accepted Answer

To find the derivative of $g(x) = \left( \frac{x + 6}{x^2 + 7} \right)^4$, we use the chain rule and the quotient rule. The derivative of the inside function $\frac{x + 6}{x^2 + 7}$ with respect to $x$ is $\frac{(1)(x^2 + 7) - (x + 6)(2x)}{(x^2 + 7)^2}$, which simplifies to $\frac{7 - 12x - x^2}{(x^2 + 7)^2}$. Applying the chain rule, we multiply this by the derivative of the outside function, which is $4\left(\frac{x + 6}{x^2 + 7}\right)^3$, resulting in the correct answer: $\frac{4(7 - 12x - x^2)(x + 6)^3}{(x^2 + 7)^5}$.

Find the Derivative of the Function g(x)=(x+6x2+7)4g ( x ) = \left( \frac { x + 6 } { x ^ { 2 } + 7 } \right) ^ { 4 }g(x)=(x2+7x+6​)4

Find the Derivative of the Function $g ( x ) = \left( \frac { x + 6 } { x ^ { 2 } + 7 } \right) ^ { 4 }$