# Differentiate Each of the Following from First Principle:$A^\Sqrt{X}$ - Mathematics

Differentiate each of the following from first principle:

$a^\sqrt{x}$

#### Solution

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


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 3.11 | Page 26