$\text { Use logarithmic differentiation to find } \frac { d y } { d x } \text {. }$ $y = \frac { 5 x - 3 } { ( 3 x + 2 ) ^ { 3 } }$
A) $\frac { - 30 x + 37 } { ( 3 x + 2 ) ^ { 5 } }$
B) $\frac { - 30 x - 17 } { ( 3 x + 2 ) ^ { 5 } }$
C) $\frac { 60 x + 37 } { ( 3 x + 2 ) ^ { 4 } }$
D) $\frac { - 30 x - 17 } { ( 3 x + 2 ) ^ { 2 } }$
E) $\frac { - 30 x + 37 } { ( 3 x + 2 ) ^ { 4 } }$

Question

Quizplus · Accepted Answer

Starting with the given function $y=\frac{5x-3}{(3x+2)^3}$, take logarithmic differentiation of both sides with respect to $x$:
\begin{align*}
\ln y &= \ln \left(\frac{5x-3}{(3x+2)^3}ight) \
&= \ln (5x-3) - 3\ln(3x+2)
\end{align*}
Now differentiate both sides implicitly with respect to $x$ using the chain rule:
\begin{align*}
\frac{1}{y} \cdot \frac{dy}{dx} &= \frac{5}{5x-3} - 3\cdot\frac{3}{3x+2} \
\frac{dy}{dx} &= y\left(\frac{5}{5x-3} - \frac{9}{3x+2}ight) \
&= \frac{5x-3}{(3x+2)^3} \cdot \left(\frac{5}{5x-3} - \frac{9}{3x+2}ight) \
&= \frac{(5x-3)(-30x+37)}{(3x+2)^5}
\end{align*}
Therefore, the correct answer is (E).

Use logarithmic differentiation to find dydx. \text { Use logarithmic differentiation to find } \frac { d y } { d x } \text {. } Use logarithmic differentiation to find dxdy​.

$\text { Use logarithmic differentiation to find } \frac { d y } { d x } \text {. }$