Question

Find the differentiation of f(x) = x².e^x at x = 1

1.  2e
2. 3e
3. e
4. None of these

Answer 1

Correct Answer - Option 2 : 3e

Concept:

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

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

\(\rm \frac{\mathrm{d} (e^{x})}{\mathrm{d} x} = e^{x}\)

Calculation:

Given: f(x) = x2.ex

Now, f'(x) = \(\rm \frac{\mathrm{d} (x^{2}e^{x})}{\mathrm{d} x}\)

By chain rule ,

⇒ \(\rm \frac{\mathrm{d} (x^{2}e^{x})}{\mathrm{d} x} = e^{x}\frac{\mathrm{d} (x^{2})}{\mathrm{d} x} + x^{2} \frac{\mathrm{d} (e^{x})}{\mathrm{d} x}\)

= ex2x +x2  ex

= xe^x(x + 2)

At x = 1;

f'(x)|_x=1 = 3e