Differentiate the function.
$$y = \ln \left( x ^ { 3 } \sin ^ { 2 } x \right)$$
A)
$y ^ { t } = \frac { 3 \cos x + 2 x \sin x } { x \cos x }$
B)
$y ^ { t } = \frac { 3 x ^ { 2 } + \cos x \sin x } { \cos x }$
C)
$y ^ { t } = \frac { 3 \sin x + x \cos x } { x ^ { 3 } \sin ^ { 2 } x }$
D)
$y ^ { t } = \frac { 3 \sin x - 2 x } { x \sin x }$
E)
$y ^ { t } = \frac { 3 \sin x + 2 x \cos x } { x \sin x }$

Question

Quizplus · Accepted Answer

We will use the chain rule to differentiate the function:
\begin{align*}
y &= \ln \left( x ^ { 3 } \sin ^ { 2 } x ight) \
y' &= \frac{1}{x^3\sin^2x}\cdot \frac{d}{dx} \left(x^3\sin^2xight) \
&= \frac{1}{x^3\sin^2x}\cdot \left(3x^2\sin^2x+x^3\cdot2\sin x\cos xight) \
&= \frac{3\sin^2x + 2x\cos x}{x\sin^3x}\
&= \frac{3\sin x + 2x\cos x}{x\sin^2x}
\end{align*}
Therefore, the correct choice is $\boxed{	extbf{(E)}}$.

Differentiate the Function y=ln⁡(x3sin⁡2x)y = \ln \left( x ^ { 3 } \sin ^ { 2 } x \right)y=ln(x3sin2x)

Differentiate the Function $y = \ln \left( x ^ { 3 } \sin ^ { 2 } x \right)$