# Differentiate Each of the Following from First Principle: E √ a X + B - Mathematics

Differentiate each of the following from first principle:

$e^\sqrt{ax + b}$

#### Solution

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

Concept: The Concept of Derivative - Algebra of Derivative of Functions
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#### APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 30 Derivatives
Exercise 30.2 | Q 3.1 | Page 26