\begin{tabular}{l} Find the derivative of $$ f(x) $$. \ $$ f(x)=x \sin (x) $$ \ $$ f^{\prime}(x)= $$\begin{tabular}{|l|l|l|l|}\hline \hline \ \hline & $$ \log _{\square} $$ & $$ \ln $$ & $$ \sin $$ \ \hline & $$ \cos $$ \ \hline $$ an $$ & $$ \csc $$ & $$ \sec $$ & $$ \cot $$ \ \hline\end{tabular} \ \hline\end{tabular}

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\begin{tabular}{l} Find the derivative of $$ f(x) $$. \ $$ f(x)=x \sin (x) $$ \ $$ f^{\prime}(x)= $$\begin{tabular}{|l|l|l|l|}\hline \hline \ \hline & $$ \log _{\square} $$ & $$ \ln $$ & $$ \sin $$ \ \hline & $$ \cos $$ \ \hline $$ 	an $$ & $$ \csc $$ & $$ \sec $$ & $$ \cot $$ \ \hline\end{tabular} \ \hline\end{tabular}

UpStudy AI · Accepted Answer

To find the derivative of the function $$ f(x) = x \sin(x) $$, we'll use the **product rule**. The product rule states that if you have a function $$ f(x) = u(x) \cdot v(x) $$, then its derivative is:

$$
f'(x) = u'(x) \cdot v(x) + u(x) \cdot v'(x)
$$

**Applying the Product Rule:**

1. Let $$ u(x) = x $$ and $$ v(x) = \sin(x) $$.
2. Compute the derivatives:
   - $$ u'(x) = 1 $$
   - $$ v'(x) = \cos(x) $$
3. Apply the product rule:
   $$
   f'(x) = u'(x) \cdot v(x) + u(x) \cdot v'(x) = 1 \cdot \sin(x) + x \cdot \cos(x)
   $$
4. Simplify:
   $$
   f'(x) = \sin(x) + x \cos(x)
   $$

**Final Answer:**

$$
f^{\prime}(x) = \sin(x) + x \cos(x)
$$
The derivative of $$ f(x) = x \sin(x) $$ is $$ f^{\prime}(x) = \sin(x) + x \cos(x) $$.

Find the derivative of .

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