# 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
Is there an error in this question or solution?

#### APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 30 Derivatives
Exercise 30.2 | Q 3.11 | Page 26
The text passage contains a mathematical question asking for the differentiation of a function from first principles, and it provides a detailed solution to the problem.
single_question_with_answer
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
# 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
Is there an error in this question or solution?

#### APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 30 Derivatives
Exercise 30.2 | Q 3.11 | Page 26
user
The text passage contains a mathematical question asking for the differentiation of a function from first principles, and it provides a detailed solution to the problem.

TYPE: SINGLE_QUESTION_WITH_ANSWER