`d/(dx)(xsinx)=……………………. .`

Answer

To find the derivative of the function \( y = x \sin x \), we will use the product rule of differentiation. The product rule states that if you have two functions \( u \) and \( v \), then the derivative of their product is given by:

\[ \frac{d}{dx}(uv) = u \frac{dv}{dx} + v \frac{du}{dx} \]

In our case, let:
- \( u = x \) (which is a function of \( x \))
- \( v = \sin x \) (which is also a function of \( x \))

Now, we will differentiate \( u \) and \( v \):

1. **Differentiate \( u \)**:
\[ \frac{du}{dx} = \frac{d}{dx}(x) = 1 \]

2. **Differentiate \( v \)**:
\[ \frac{dv}{dx} = \frac{d}{dx}(\sin x) = \cos x \]

Now, applying the product rule:

\[ \frac{d}{dx}(x \sin x) = u \frac{dv}{dx} + v \frac{du}{dx} \]

Substituting the values we have:

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

This simplifies to:

\[ \frac{d}{dx}(x \sin x) = x \cos x + \sin x \]

Thus, the final answer is:

\[ \frac{d}{dx}(x \sin x) = \sin x + x \cos x \]

---