text: # Thread: Trouble with Quotient Rule
  
  1. ## Trouble with Quotient Rule
  
  I need to simplify it further, but before I do I checked it into Mathematica and it says the derivative is incorrect. What did I do wrong?
  
  2. ## Re: Trouble with Quotient Rule
  
  Your calculations are correct. If you simplify you should get
  1/(x^2 sqrt(1+x^2))
  
  3. ## Re: Trouble with Quotient Rule
  
  I find it to be a bit easier, though there is more factoring involved, if you divide by x first:
  $\frac{d}{dx} \left ( - \frac{\sqrt{1 + x^2}}{x} \right ) = -\frac{d}{dx} \left ( \sqrt{ \frac{1}{x^2} + 1} \right )$
  
  $= - \frac{d}{dx} \left ( \left [ \frac{1}{x^2} + 1 \right ] ^{1/2} \right )$
  
  Now for the derivative:
  $= -\frac{1}{2} \cdot \left ( \frac{1}{x^2} + 1 \right ) ^{-1/2} \cdot \frac{-2}{x^3}$
  
  $= \frac{1}{x^3 \left ( \frac{1}{x^2} + 1 \right ) ^{1/2} }$
  
  I find this result to be reasonably adequate, but your text decided differently and cleared the fraction in the denominator. So, using a little trick:
  $= \frac{1}{x^2 \cdot x \left ( \frac{1}{x^2} + 1 \right ) ^{1/2} }$
  
  $= \frac{1}{x^2 \left ( x^2 \cdot \frac{1}{x^2} + x^2 \cdot 1 \right ) ^{1/2} }$
  
  $= \frac{1}{x^2 \left ( 1 + x^2 \right ) ^{1/2} }$
  
  -Dan
explanation: The document is a discussion thread about a math problem involving the quotient rule. The initial post presents a question about an incorrect derivative, and subsequent posts provide an answer and an alternative method to solve the problem.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: # Thread: Trouble with Quotient Rule
  
  1. ## Trouble with Quotient Rule
  
  I need to simplify it further, but before I do I checked it into Mathematica and it says the derivative is incorrect. What did I do wrong?
  
  2. ## Re: Trouble with Quotient Rule
  
  Your calculations are correct. If you simplify you should get
  1/(x^2 sqrt(1+x^2))
  
  3. ## Re: Trouble with Quotient Rule
  
  I find it to be a bit easier, though there is more factoring involved, if you divide by x first:
  $\frac{d}{dx} \left ( - \frac{\sqrt{1 + x^2}}{x} \right ) = -\frac{d}{dx} \left ( \sqrt{ \frac{1}{x^2} + 1} \right )$
  
  $= - \frac{d}{dx} \left ( \left [ \frac{1}{x^2} + 1 \right ] ^{1/2} \right )$
  
  Now for the derivative:
  $= -\frac{1}{2} \cdot \left ( \frac{1}{x^2} + 1 \right ) ^{-1/2} \cdot \frac{-2}{x^3}$
  
  $= \frac{1}{x^3 \left ( \frac{1}{x^2} + 1 \right ) ^{1/2} }$
  
  I find this result to be reasonably adequate, but your text decided differently and cleared the fraction in the denominator. So, using a little trick:
  $= \frac{1}{x^2 \cdot x \left ( \frac{1}{x^2} + 1 \right ) ^{1/2} }$
  
  $= \frac{1}{x^2 \left ( x^2 \cdot \frac{1}{x^2} + x^2 \cdot 1 \right ) ^{1/2} }$
  
  $= \frac{1}{x^2 \left ( 1 + x^2 \right ) ^{1/2} }$
  
  -Dan
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document is a discussion thread about a math problem involving the quotient rule. The initial post presents a question about an incorrect derivative, and subsequent posts provide an answer and an alternative method to solve the problem.
  
  Label: question-and-answer