Calculate $\frac { \mathrm { d } y } { \mathrm {~d} x }$ . You need not expand your answer. $y = \frac { 3 x - 6 } { ( x - 3 ) ( x - 1 ) ( x - 3 ) }$
A) $\frac { 3 ( x - 3 ) ( x - 1 ) ( x - 3 ) - \left( 3 x ^ { 2 } - 14 x + 15 \right) ( 3 x - 6 ) } { ( ( x - 3 ) ( x - 1 ) ( x - 3 ) ) ^ { 2 } }$
B) $\frac { 3 ( x - 3 ) ( x - 1 ) ( x - 3 ) - \left( 3 x ^ { 2 } - 14 x + 15 \right) ( 3 x - 6 ) } { ( x - 3 ) ( x - 1 ) ( x - 3 ) }$
C) $\frac { 3 ( x - 3 ) ( x - 1 ) ( x - 3 ) + \left( 3 x ^ { 2 } - 14 x + 7 \right) ( 3 x - 6 ) } { ( ( x - 3 ) ( x - 1 ) ( x - 3 ) ) ^ { 2 } }$
D) $\frac { 3 ( x - 3 ) ( x - 3 ) - \left( 3 x ^ { 2 } - 14 x + 15 \right) } { ( ( x - 3 ) ( x - 3 ) ) ^ { 2 } }$
E) $\frac { 3 } { 3 x ^ { 2 } - 14 x + 7 }$

Question

Calculate $\frac { \mathrm { d } y } { \mathrm {~d} x }$ . You need not expand your answer.   $y = \frac { 3 x - 6 } { ( x - 3 ) ( x - 1 ) ( x - 3 ) }$  
A) $\frac { 3 ( x - 3 ) ( x - 1 ) ( x - 3 ) - \left( 3 x ^ { 2 } - 14 x + 15 \right) ( 3 x - 6 ) } { ( ( x - 3 ) ( x - 1 ) ( x - 3 ) ) ^ { 2 } }$
B) $\frac { 3 ( x - 3 ) ( x - 1 ) ( x - 3 ) - \left( 3 x ^ { 2 } - 14 x + 15 \right) ( 3 x - 6 ) } { ( x - 3 ) ( x - 1 ) ( x - 3 ) }$
C) $\frac { 3 ( x - 3 ) ( x - 1 ) ( x - 3 ) + \left( 3 x ^ { 2 } - 14 x + 7 \right) ( 3 x - 6 ) } { ( ( x - 3 ) ( x - 1 ) ( x - 3 ) ) ^ { 2 } }$
D) $\frac { 3 ( x - 3 ) ( x - 3 ) - \left( 3 x ^ { 2 } - 14 x + 15 \right) } { ( ( x - 3 ) ( x - 3 ) ) ^ { 2 } }$
E) $\frac { 3 } { 3 x ^ { 2 } - 14 x + 7 }$

Quizplus · Accepted Answer

The correct answer involves applying the quotient rule for differentiation, which is given by $\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v\frac{du}{dx} - u\frac{dv}{dx}}{v^2}$, where $u = 3x - 6$ and $v = (x - 3)^2(x - 1)$. The differentiation of the numerator and denominator and their substitution into the quotient rule formula results in the expression given in option A.

Calculate dy dx\frac { \mathrm { d } y } { \mathrm {~d} x } dxdy​

Calculate $\frac { \mathrm { d } y } { \mathrm {~d} x }$