text: # Quotient Rule
  
  How to differentiate tanx using the quotient rule
  
  Tip: This isn't the place to ask a question because the teacher can't reply.
  
  1 of 2 videos by Tiago Hands
  
  ## Key Questions
  
  • $y ' = \frac{g \left(x\right) f ' \left(x\right) - f \left(x\right) g ' \left(x\right)}{g \left(x\right)} ^ 2$
  
  $y = f \frac{x}{g} \left(x\right) = \frac{2 {x}^{4} - 3 x}{4 x - 1}$
  
  $f ' \left(x\right) = 8 {x}^{3} - 3$
  
  $g ' \left(x\right) = 4$
  
  ${\left(g \left(x\right)\right)}^{2} = {\left(4 x - 1\right)}^{2}$
  
  $y ' = \frac{\left(4 x - 1\right) \left(8 {x}^{3} - 3\right) - \left(2 {x}^{4} - 3 x\right) \left(4\right)}{4 x - 1} ^ 2$
  
  $y ' = \frac{32 {x}^{4} - 12 x - 8 {x}^{3} + 3 - 8 {x}^{4} + 12 x}{4 x - 1} ^ 2$
  
  Simplify for combining like terms.
  
  $S o l u t i o n \to y ' = \frac{24 {x}^{4} - 8 {x}^{3} + 3}{4 x - 1} ^ 2$
  
  • $y ' = \frac{1}{\sqrt{x}} \cdot \frac{1}{{\left(1 - \sqrt{x}\right)}^{2}}$
  
  Explanation :
  
  Using Quotient Rule, which is
  
  $y = f \frac{x}{g} \left(x\right)$, then
  
  $y ' = \frac{g \left(x\right) f ' \left(x\right) - f \left(x\right) g ' \left(x\right)}{g \left(x\right)} ^ 2$
  
  Similarly following for the given problem,
  
  $y = \frac{1 + \sqrt{x}}{1 - \sqrt{x}}$
  
  $y ' = \frac{\left(1 - \sqrt{x}\right) \left(\frac{1}{2 \sqrt{x}}\right) - \left(1 + \sqrt{x}\right) \left(- \frac{1}{2 \sqrt{x}}\right)}{{\left(1 - \sqrt{x}\right)}^{2}}$
  
  $y ' = \frac{1}{2 \sqrt{x}} \cdot \frac{1 - \sqrt{x} + 1 + \sqrt{x}}{{\left(1 - \sqrt{x}\right)}^{2}}$
  
  $y ' = \frac{1}{2 \sqrt{x}} \cdot \frac{2}{{\left(1 - \sqrt{x}\right)}^{2}}$
  
  $y ' = \frac{1}{\sqrt{x}} \cdot \frac{1}{{\left(1 - \sqrt{x}\right)}^{2}}$
  
  • This key question hasn't been answered yet.
  
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explanation: The document appears to be a set of examples and explanations related to the quotient rule in calculus. It includes several questions and their corresponding answers, demonstrating how to apply the quotient rule to differentiate various functions. The document also contains a tip about not asking questions in a specific context, likely referring to a platform where the content is hosted.
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: # Quotient Rule
  
  How to differentiate tanx using the quotient rule
  
  Tip: This isn't the place to ask a question because the teacher can't reply.
  
  1 of 2 videos by Tiago Hands
  
  ## Key Questions
  
  • $y ' = \frac{g \left(x\right) f ' \left(x\right) - f \left(x\right) g ' \left(x\right)}{g \left(x\right)} ^ 2$
  
  $y = f \frac{x}{g} \left(x\right) = \frac{2 {x}^{4} - 3 x}{4 x - 1}$
  
  $f ' \left(x\right) = 8 {x}^{3} - 3$
  
  $g ' \left(x\right) = 4$
  
  ${\left(g \left(x\right)\right)}^{2} = {\left(4 x - 1\right)}^{2}$
  
  $y ' = \frac{\left(4 x - 1\right) \left(8 {x}^{3} - 3\right) - \left(2 {x}^{4} - 3 x\right) \left(4\right)}{4 x - 1} ^ 2$
  
  $y ' = \frac{32 {x}^{4} - 12 x - 8 {x}^{3} + 3 - 8 {x}^{4} + 12 x}{4 x - 1} ^ 2$
  
  Simplify for combining like terms.
  
  $S o l u t i o n \to y ' = \frac{24 {x}^{4} - 8 {x}^{3} + 3}{4 x - 1} ^ 2$
  
  • $y ' = \frac{1}{\sqrt{x}} \cdot \frac{1}{{\left(1 - \sqrt{x}\right)}^{2}}$
  
  Explanation :
  
  Using Quotient Rule, which is
  
  $y = f \frac{x}{g} \left(x\right)$, then
  
  $y ' = \frac{g \left(x\right) f ' \left(x\right) - f \left(x\right) g ' \left(x\right)}{g \left(x\right)} ^ 2$
  
  Similarly following for the given problem,
  
  $y = \frac{1 + \sqrt{x}}{1 - \sqrt{x}}$
  
  $y ' = \frac{\left(1 - \sqrt{x}\right) \left(\frac{1}{2 \sqrt{x}}\right) - \left(1 + \sqrt{x}\right) \left(- \frac{1}{2 \sqrt{x}}\right)}{{\left(1 - \sqrt{x}\right)}^{2}}$
  
  $y ' = \frac{1}{2 \sqrt{x}} \cdot \frac{1 - \sqrt{x} + 1 + \sqrt{x}}{{\left(1 - \sqrt{x}\right)}^{2}}$
  
  $y ' = \frac{1}{2 \sqrt{x}} \cdot \frac{2}{{\left(1 - \sqrt{x}\right)}^{2}}$
  
  $y ' = \frac{1}{\sqrt{x}} \cdot \frac{1}{{\left(1 - \sqrt{x}\right)}^{2}}$
  
  • This key question hasn't been answered yet.
  
  ## Questions
  
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distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document appears to be a set of examples and explanations related to the quotient rule in calculus. It includes several questions and their corresponding answers, demonstrating how to apply the quotient rule to differentiate various functions. The document also contains a tip about not asking questions in a specific context, likely referring to a platform where the content is hosted.
  
  Label: multiple-question-and-answer