$\text { Use the Quotient Rule to differentiate the function } f ( x ) = \frac { 8 x } { x ^ { 5 } + 3 } \text {. }$
A)$f ^ { t} ( x ) = - \frac { 8 \left( - 3 + 4 x ^ { 5 } \right) } { \left( x ^ { 5 } + 3 \right) ^ { 2 } }$
B)$f ^ { t } ( x ) = \frac { 8 \left( - 3 - 4 x ^ { 5 } \right) } { \left( x ^ { 5 } + 3 \right) ^ { 2 } }$
C)$f ^ { t } ( x ) = - \frac { 8 \left( 3 + 5 x ^ { 5 } \right) } { \left( x ^ { 5 } + 3 \right) ^ { 2 } }$
D)$f ^ { t } ( x ) = \frac { 8 \left( 3 + 4 x ^ { 5 } \right) } { \left( x ^ { 5 } + 3 \right) ^ { 2 } }$
E)$f ^ { t } ( x ) = - \frac { 8 \left( 3 + 6 x ^ { 5 } \right) } { \left( x ^ { 5 } + 3 \right) ^ { 2 } }$

Question

$\text { Use the Quotient Rule to differentiate the function } f ( x ) = \frac { 8 x } { x ^ { 5 } + 3 } \text {. }$ 
A)$f ^ { t} ( x ) = - \frac { 8 \left( - 3 + 4 x ^ { 5 } \right) } { \left( x ^ { 5 } + 3 \right) ^ { 2 } }$
B)$f ^ { t } ( x ) = \frac { 8 \left( - 3 - 4 x ^ { 5 } \right) } { \left( x ^ { 5 } + 3 \right) ^ { 2 } }$
C)$f ^ { t } ( x ) = - \frac { 8 \left( 3 + 5 x ^ { 5 } \right) } { \left( x ^ { 5 } + 3 \right) ^ { 2 } }$
D)$f ^ { t } ( x ) = \frac { 8 \left( 3 + 4 x ^ { 5 } \right) } { \left( x ^ { 5 } + 3 \right) ^ { 2 } }$
E)$f ^ { t } ( x ) = - \frac { 8 \left( 3 + 6 x ^ { 5 } \right) } { \left( x ^ { 5 } + 3 \right) ^ { 2 } }$

Quizplus · Accepted Answer

The Quotient Rule for differentiation is given by $\frac{d}{dx} \left( \frac{u}{v} \right) = \frac{v(u') - u(v')}{v^2}$, where $u = 8x$ and $v = x^5 + 3$. Differentiating $u$ and $v$ gives $u' = 8$ and $v' = 5x^4$. Substituting these into the Quotient Rule formula yields $f'(x) = \frac{(x^5 + 3)(8) - (8x)(5x^4)}{(x^5 + 3)^2}$, which simplifies to $- \frac { 8 \left( - 3 + 4 x ^ { 5 } \right) } { \left( x ^ { 5 } + 3 \right) ^ { 2 } }$.

Use the Quotient Rule to differentiate the function f(x)=8xx5+3. \text { Use the Quotient Rule to differentiate the function } f ( x ) = \frac { 8 x } { x ^ { 5 } + 3 } \text {. } Use the Quotient Rule to differentiate the function f(x)=x5+38x​.

$\text { Use the Quotient Rule to differentiate the function } f ( x ) = \frac { 8 x } { x ^ { 5 } + 3 } \text {. }$