text: # 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
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distilabel_metadata.content: # 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
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distilabel_metadata.raw_output_q_a_detector_0: The document contains a math question and its solution. The question asks to differentiate \( a^\sqrt{x} \) from first principles, and the solution provides a step-by-step derivation of the derivative.
  
  Label: question-and-answer