To find \(\frac{dy}{dx}\) for the function \(y = x \sin(x)\), we'll need to apply the product rule of differentiation. The product rule states that if \(u(x)\) and \(v(x)\) are two functions of \(x\), then the derivative of their product is given by:

\(\frac{d}{dx} (u(x) \cdot v(x)) = u'(x) \cdot v(x) + u(x) \cdot v'(x)\(

In this case, \(u(x) = x\) and \(v(x) = \sin(x)\). Taking the derivatives:

\(u'(x) = 1\(

\(v'(x) = \cos(x)\(

Applying the product rule:

\(\frac{d}{dx} (x \sin(x)) = 1 \cdot \sin(x) + x \cdot \cos(x)\(

So, \(\frac{dy}{dx} = \sin(x) + x \cos(x)\),