$\text { Use the Quotient Rule to differentiate the function } f ( x ) = \frac { \sin x } { x ^ { 2 } + 3 } \text {. }$
A)
$$f ^ {t } ( x ) = \frac { \left( 3 + x ^ { 2 } \right) \cos x + 2 x \sin x } { \left( x ^ { 2 } + 3 \right) ^ { 2 } }$$
B)
$$f ^ { t } ( x ) = \frac { \left( 3 + x ^ { 2 } \right) \cos x - 2 x \sin x } { \left( x ^ { 2 } + 3 \right) ^ { 2 } }$$
C)
$$f ^ { t } ( x ) = \frac { ( 3 + x ) \cos x - 2 x \sin x } { \left( x ^ { 2 } + 3 \right) ^ { 2 } }$$
D)
$$f ^ { t } ( x ) = \frac { \left( 3 - x ^ { 2 } \right) \cos x - 2 x \sin x } { \left( x ^ { 2 } + 3 \right) ^ { 2 } }$$
E)
$$f ^ { t} ( x ) = \frac { \left( 3 + x ^ { 2 } \right) \cos x - 2 x \sin x } { \left( x ^ { 2 } + 3 \right) }$$

Question

$\text { Use the Quotient Rule to differentiate the function } f ( x ) = \frac { \sin x } { x ^ { 2 } + 3 } \text {. }$ 
A)
$$f ^ {t } ( x ) = \frac { \left( 3 + x ^ { 2 } \right) \cos x + 2 x \sin x } { \left( x ^ { 2 } + 3 \right) ^ { 2 } }$$
B)
$$f ^ { t } ( x ) = \frac { \left( 3 + x ^ { 2 } \right) \cos x - 2 x \sin x } { \left( x ^ { 2 } + 3 \right) ^ { 2 } }$$
C)
$$f ^ { t } ( x ) = \frac { ( 3 + x ) \cos x - 2 x \sin x } { \left( x ^ { 2 } + 3 \right) ^ { 2 } }$$
D)
$$f ^ { t } ( x ) = \frac { \left( 3 - x ^ { 2 } \right) \cos x - 2 x \sin x } { \left( x ^ { 2 } + 3 \right) ^ { 2 } }$$
E)
$$f ^ { t} ( x ) = \frac { \left( 3 + x ^ { 2 } \right) \cos x - 2 x \sin x } { \left( x ^ { 2 } + 3 \right) }$$

Quizplus · Accepted Answer

Applying the quotient rule, we get
\begin{align*}
f'(x) &= \frac{(x^2+3)\cos x \cdot 1 - \sin x \cdot 2x}{(x^2+3)^2} \
&= \frac{(3+x^2)\cos x-2x\sin x}{(x^2+3)^2}
\end{align*} 
Therefore, the correct choice is (B).

Use the Quotient Rule to differentiate the function f(x)=sin⁡xx2+3. \text { Use the Quotient Rule to differentiate the function } f ( x ) = \frac { \sin x } { x ^ { 2 } + 3 } \text {. } Use the Quotient Rule to differentiate the function f(x)=x2+3sinx​.

$\text { Use the Quotient Rule to differentiate the function } f ( x ) = \frac { \sin x } { x ^ { 2 } + 3 } \text {. }$