If y = e^xsin x , then displaystyledy/dx is equal to

The correct answer is Option (3) → \((\sin x + x\cos x)\, e^{x\sin x}\)

Let \(u = x\sin x\), so \(y = e^{u}\). By the chain rule:

    \( \displaystyle \frac{dy}{dx} = e^{u} \cdot \frac{du}{dx} \)

By the product rule:

    \( \displaystyle \frac{du}{dx} = \sin x + x\cos x \)

So

    \( \displaystyle \frac{dy}{dx} = (\sin x + x\cos x)\, e^{x\sin x} \)