Find the derivative of the function.
-$$g ( x ) = \frac { x ^ { 2 } + 5 } { x ^ { 2 } + 6 x }$$
A) $g ^ { \prime } ( x ) = \frac { 2 x ^ { 3 } - 5 x ^ { 2 } - 30 x } { x ^ { 2 } ( x + 6 ) ^ { 2 } }$
B) $g ^ { \prime } ( x ) = \frac { x ^ { 4 } + 6 x ^ { 3 } + 5 x ^ { 2 } + 30 x } { x ^ { 2 } ( x + 6 ) ^ { 2 } }$
C) $g ^ { \prime } ( x ) = \frac { 4 x ^ { 3 } + 18 x ^ { 2 } + 10 x + 30 } { x ^ { 2 } ( x + 6 ) ^ { 2 } }$
D) $g ^ { \prime } ( x ) = \frac { 6 x ^ { 2 } - 10 x - 30 } { x ^ { 2 } ( x + 6 ) ^ { 2 } }$

Question

Quizplus · Accepted Answer

We can use the quotient rule to find the derivative of the function $g(x):$

\begin{align*}
g'(x) &= \frac{(x^2+6x)(2x)-(x^2+5)(2x+6)}{(x^2+6x)^2} \
&= \frac{2x^3+12x^2-2x^3-10x-30}{(x^2+6x)^2} \
&= \frac{-10x + 12x^2}{(x^2+6x)^2} \
&= \frac{2x(6-x)}{x^2(x+6)^2} \
&= \frac{2(6-x)}{x^2(x+6)}.
\end{align*}

We can simplify the result as follows:

\begin{align*}
g'(x) &= \frac{2(6-x)}{x^2(x+6)} \
&= \frac{2(-x+6)}{x^2(x+6)} \
&= \frac{-2x + 12}{x^2(x+6)} \
&= \frac{6-2x-18}{x^2(x+6)} \
&= \frac{6}{x^2(x+6)} - \frac{2x+18}{x^2(x+6)} \
&= \frac{6}{x^2(x+6)} - \frac{2(x+6)}{x^2(x+6)} \
&= \frac{6-2(x+6)}{x^2(x+6)} \
&= \frac{6-2x-12}{x^2(x+6)} \
&= \frac{-2x-6}{x^2(x+6)} \
&= \frac{-2(x+3)}{x^2(x+6)} \
&= \frac{6(x+3)}{-3x^2(x+6)} \
&= \frac{6(3+x)}{3x^2(x+6)} \
&= \frac{6(3+x)}{(x^2+6x)^2}.
\end{align*}

Therefore, the answer is $\boxed{	extbf{(D)}}.$

Find the Derivative of the Function g(x)=x2+5x2+6xg ( x ) = \frac { x ^ { 2 } + 5 } { x ^ { 2 } + 6 x }g(x)=x2+6xx2+5​

Find the Derivative of the Function $g ( x ) = \frac { x ^ { 2 } + 5 } { x ^ { 2 } + 6 x }$