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

# Differentiate Each of the Following from First Principle: E √ 2 X - Mathematics

Differentiate each  of the following from first principle:

$e^\sqrt{2x}$

#### Solution

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

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#### APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 30 Derivatives
Exercise 30.2 | Q 3.09 | Page 26