\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 \( \tan \) & \( \csc \) & \( \sec \) & \( \cot \) \\ \hline\end{tabular} \\ \hline\end{tabular}

To find the derivative of \( f(x) = x \sin(x) \), we can apply the product rule, which states that if you have two functions \( u(x) \) and \( v(x) \), then the derivative of their product is given by \( (uv)' = u'v + uv' \). Here, let \( u(x) = x \) and \( v(x) = \sin(x) \). First, we find the derivatives: - \( u'(x) = 1 \) - \( v'(x) = \cos(x) \) Now, applying the product rule: \[ f'(x) = u'v + uv' = 1 \cdot \sin(x) + x \cdot \cos(x) = \sin(x) + x \cos(x) \] So, the derivative is: \[ f'(x) = \sin(x) + x \cos(x) \] Now, let’s enrich your understanding a bit! Did you know that the product rule is one of the fundamental tools in calculus? It allows you to differentiate products of functions, which is essential when dealing with physics equations, economics models, and even in computer algorithms where you might be working with rates of change or optimizing functions. If you want to delve deeper into the beauty of calculus, consider exploring textbooks by authors like James Stewart or online resources like Khan Academy, which offer interactive lessons and problems tailored to various skill levels, ensuring you grasp the concept of derivatives thoroughly!