Question

If f(x) = x³sinx, Find f'(x)

1. (x cos x + 3sin x)
2. x2sin x)
3. x2(x cos x + 3sin x)
4. x2 cos x

Answer 1

Correct Answer - Option 3 : x2(x cos x + 3sin x)

Concept:

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

\(\rm \frac{\mathrm{d} (uv)}{\mathrm{d} x} = u\frac{\mathrm{d} v}{\mathrm{d} x} + v \frac{\mathrm{d} u}{\mathrm{d} x}\)

\(\rm \frac{\mathrm{d} (\sin x)}{\mathrm{d} x} = \cos x\)

Calculation:

Given: f(x) = x3sinx

f'(x) = \(\rm \frac{\mathrm{d} (x^{3}sinx)}{\mathrm{d} x}\)

⇒ \(\rm \frac{\mathrm{d} (x^{3}sinx)}{\mathrm{d} x} = x^{3}\frac{\mathrm{d} (sinx)}{\mathrm{d} x} + sinx \frac{\mathrm{d} x^{3}}{\mathrm{d} x}\)

= x³ cos x + sin x 3x²

= x²(x cos x + 3sin x)

∴ The required value is x2(x cos x + 3sin x).