Find `dy/dx if y= x sinx`

Answer

To find \(\frac{dy}{dx}\) for 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 \(uv\) is given by:

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

In our case, we can identify:
- \(u = x\)
- \(v = \sin x\)

Now, we will differentiate each part:

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:

\[ \frac{dy}{dx} = u \frac{dv}{dx} + v \frac{du}{dx} \]

Substituting the values we found:

\[ \frac{dy}{dx} = x \cdot \cos x + \sin x \cdot 1 \]

This simplifies to:

\[ \frac{dy}{dx} = x \cos x + \sin x \]

Thus, the final answer is:

\[ \frac{dy}{dx} = x \cos x + \sin x \]

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