text: If $$y = ( tan^{-1}x )^2$$, show that $$(x^2 + 1 )^2 \: y_2 + 2x (x^2 +1 ) \: y_1 = 2$$
  
  Toolbox:
  • $y=f(x)$
  • $\large\frac{dy}{dx}$$=f'(x) • \large\frac{d^2y}{dx^2}=\frac{d}{dx}\big(\frac{dy}{dx}\big) • \large\frac{d}{dx}$$(\tan^{-1}x)=\large\frac{1}{1+x^2}$
  Step 1:
  We have $y=(\tan^{-1}x)^2$
  $y=(\tan^{-1}x)^2$
  $y_1=2\tan{-1}x\large\frac{d}{dx}$$(\tan^{-1}x)$
  $y_1=2\tan^{-1}x.\large\frac{1}{1+x^2}$
  $y_1=\large\frac{2\tan^{-1}x}{1+x^2}$
  $(1+x^2)y_1=2\tan^{-1}x$
  Step 2:
  Differentiating with respect to $x$ we have
  $(1+x^2)y_2+y_1(2x)=\large\frac{2}{1+x^2}$
  $(1+x^2)^2y_2+2x(1+x^2)y_1=2$
  Hence proved.
explanation: The document contains a math problem and its solution. The problem is to show that for the function \( y = (\tan^{-1}x)^2 \), the equation \( (x^2 + 1)^2 y_2 + 2x (x^2 + 1) y_1 = 2 \) holds true. The document provides a step-by-step solution to this 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: If $$y = ( tan^{-1}x )^2$$, show that $$(x^2 + 1 )^2 \: y_2 + 2x (x^2 +1 ) \: y_1 = 2$$
  
  Toolbox:
  • $y=f(x)$
  • $\large\frac{dy}{dx}$$=f'(x) • \large\frac{d^2y}{dx^2}=\frac{d}{dx}\big(\frac{dy}{dx}\big) • \large\frac{d}{dx}$$(\tan^{-1}x)=\large\frac{1}{1+x^2}$
  Step 1:
  We have $y=(\tan^{-1}x)^2$
  $y=(\tan^{-1}x)^2$
  $y_1=2\tan{-1}x\large\frac{d}{dx}$$(\tan^{-1}x)$
  $y_1=2\tan^{-1}x.\large\frac{1}{1+x^2}$
  $y_1=\large\frac{2\tan^{-1}x}{1+x^2}$
  $(1+x^2)y_1=2\tan^{-1}x$
  Step 2:
  Differentiating with respect to $x$ we have
  $(1+x^2)y_2+y_1(2x)=\large\frac{2}{1+x^2}$
  $(1+x^2)^2y_2+2x(1+x^2)y_1=2$
  Hence proved.
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document contains a math problem and its solution. The problem is to show that for the function \( y = (\tan^{-1}x)^2 \), the equation \( (x^2 + 1)^2 y_2 + 2x (x^2 + 1) y_1 = 2 \) holds true. The document provides a step-by-step solution to this problem.
  
  Label: question-and-answer