Department of Pre-University Education, KarnatakaPUC Karnataka Science Class 11

# Differentiate Each of the Following from First Principle: X2 Ex - Mathematics

Differentiate each of the following from first principle:

x2 e

#### Solution

$\frac{d}{dx}\left( f(x) \right) = \lim_{h \to 0} \frac{f\left( x + h \right) - f\left( x \right)}{h}$
$\frac{d}{dx}\left( x^2 e^x \right) = \lim_{h \to 0} \frac{(x + h )^2 e^{(x + h)} - x^2 e^x}{h}$
$= \lim_{h \to 0} \frac{( x^2 + 2xh + h^2 ) e^x e^h - x^2 e^x}{h}$
$= \lim_{h \to 0} \frac{x^2 e^x e^h + 2xh e^x e^h + h^2 e^x e^h - x^2 e^x}{h}$
$= \lim_{h \to 0} \frac{x^2 e^x e^h - x^2 e^x}{h} + \lim_{h \to 0} \frac{2 x h e^x e^h}{h} + \lim_{h \to 0} \frac{h^2 e^x e^h}{h}$
$= \lim_{h \to 0} \frac{x^2 e^x \left( e^h - 1 \right)}{h} + \lim_{h \to 0} 2 x e^x e^h + \lim_{h \to 0} h e^x e^h$
$= x^2 e^x \left( 1 \right) + 2x e^x \left( 1 \right) + 0$
$= x^2 e^x + 2x e^x$
$= \left( x^2 + 2x \right) e^x$

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#### APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 30 Derivatives
Exercise 30.2 | Q 3.07 | Page 26