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If y = 2x cos x + x2 sin x, then \(\rm \frac{dy}{dx}\) = ?
1. cos x + 2x sin x
2. -cos x + x2 cos x
3. x2 cos x - 2x
4. (x2 + 2) cos x

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Correct Answer - Option 4 : (x2 + 2) cos x

Concept:

Derivatives:

Product rule:

For two functions u and v of x, we have:

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

\(\rm \frac{d}{dx}x^n=nx^{n-1}\)

Calculation:

Using the product rule of derivatives, we get:

\(\rm \frac{dy}{dx}\) = \(\rm \frac{d}{dx} \left(2x \cos x + x^2 \sin x\right)\)

= (2 cos x - 2x sin x) + (2x sin x + x2 cos x)

= 2 cos x + x2 cos x

= (x2 + 2) cos x

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