$\text { Use the Quotient Rule to differentiate the function } f ^ {t } ( x ) = \frac { 4 + x } { x ^ { 2 } + 9 } \text {. }$
A) $f ^ { t } ( x ) = \frac { \left( 9 + 8 x - x ^ { 2 } \right) } { \left( x ^ { 2 } + 9 \right) ^ { 2 } }$
B) $f ^ { t } ( x ) = \frac { \left( 9 - 8 x - x ^ { 2 } \right) } { \left( x ^ { 2 } + 9 \right) ^ { 2 } }$
C)$f ^ { t} ( x ) = \frac { \left( 9 - 4 x - x ^ { 2 } \right) } { \left( x ^ { 2 } + 9 \right) ^ { 2 } }$
D)$f ^ { t } ( x ) = - \frac { \left( 9 - 8 x - x ^ { 2 } \right) } { \left( x ^ { 2 } + 9 \right) ^ { 2 } }$
E)$f ^ { t } ( x ) = \frac { \left( 9 - 8 x + x ^ { 2 } \right) } { \left( x ^ { 2 } + 9 \right) ^ { 2 } }$

Question

$\text { Use the Quotient Rule to differentiate the function } f ^ {t } ( x ) = \frac { 4 + x } { x ^ { 2 } + 9 } \text {. }$ 
A) $f ^ { t } ( x ) = \frac { \left( 9 + 8 x - x ^ { 2 } \right) } { \left( x ^ { 2 } + 9 \right) ^ { 2 } }$
B) $f ^ { t } ( x ) = \frac { \left( 9 - 8 x - x ^ { 2 } \right) } { \left( x ^ { 2 } + 9 \right) ^ { 2 } }$
C)$f ^ { t} ( x ) = \frac { \left( 9 - 4 x - x ^ { 2 } \right) } { \left( x ^ { 2 } + 9 \right) ^ { 2 } }$
D)$f ^ { t } ( x ) = - \frac { \left( 9 - 8 x - x ^ { 2 } \right) } { \left( x ^ { 2 } + 9 \right) ^ { 2 } }$
E)$f ^ { t } ( x ) = \frac { \left( 9 - 8 x + x ^ { 2 } \right) } { \left( x ^ { 2 } + 9 \right) ^ { 2 } }$

Quizplus · Accepted Answer

When using the Quotient Rule, we have $$\left(\frac{f(x)}{g(x)}\right)'=\frac{f'(x)g(x)-f(x)g'(x)}{\left(g(x)\right)^2}.$$ Applying this to $f^t(x)=\frac{4+x}{x^2+9}$, we have $$f^t(x)'=\frac{(1)(x^2+9)-(4+x)(2x)}{(x^2+9)^2}=\frac{-x^2-8x+9}{(x^2+9)^2}.$$ Therefore, the answer is B.

Use the Quotient Rule to differentiate the function ft(x)=4+xx2+9. \text { Use the Quotient Rule to differentiate the function } f ^ {t } ( x ) = \frac { 4 + x } { x ^ { 2 } + 9 } \text {. } Use the Quotient Rule to differentiate the function ft(x)=x2+94+x​.

$\text { Use the Quotient Rule to differentiate the function } f ^ {t } ( x ) = \frac { 4 + x } { x ^ { 2 } + 9 } \text {. }$