text: 1. ## Derivative Calculation
  
  Given:
  
  $\displaystyle y= \frac{sin^2 x}{1+ ctgx} +$$\displaystyle \frac{cos^2 x}{1+ ctgx}$
  
  Find $\displaystyle y'$
  
  I know that answer is $\displaystyle -cos2x$, but can't calculate it.
  
  I got so far:
  
  $\displaystyle y'=(\frac{sin^2 x}{1+ ctgx} +\frac{cos^2 x}{1+ ctgx} )'= \frac{(sin^2 x)'(1+ ctgx) - sin^2 x (1+ ctgx)'}{(1+ ctgx)^2}$ + $\displaystyle \frac{(cos^2 x)'(1+ tgx) - cos^2 x (1+ tgx)'}{(1+ tgx)^2} =$
  
  $\displaystyle =\frac{2sinx(1+ctgx)+1}{(1+ ctgx)^2} +\frac{2cos x(1+tgx)-1}{(1+ tgx)^2}$
  
  2. I spot one crucial error. There might be more but you need to correct this one first.
  
  $\displaystyle (\cos^2(x))' = -2\cos(x)\sin(x)$ and $\displaystyle (\sin^2(x))' = 2\sin(x)\cos(x)$
explanation: The document contains a math problem involving the calculation of a derivative. The user provides the given function and asks for help in finding the derivative, mentioning that they know the answer is \(-\cos 2x\) but are unable to derive it. The user also shows their work and identifies a potential error in their calculation. Another user points out a crucial error in the derivatives of \(\cos^2(x)\) and \(\sin^2(x)\).
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: 1. ## Derivative Calculation
  
  Given:
  
  $\displaystyle y= \frac{sin^2 x}{1+ ctgx} +$$\displaystyle \frac{cos^2 x}{1+ ctgx}$
  
  Find $\displaystyle y'$
  
  I know that answer is $\displaystyle -cos2x$, but can't calculate it.
  
  I got so far:
  
  $\displaystyle y'=(\frac{sin^2 x}{1+ ctgx} +\frac{cos^2 x}{1+ ctgx} )'= \frac{(sin^2 x)'(1+ ctgx) - sin^2 x (1+ ctgx)'}{(1+ ctgx)^2}$ + $\displaystyle \frac{(cos^2 x)'(1+ tgx) - cos^2 x (1+ tgx)'}{(1+ tgx)^2} =$
  
  $\displaystyle =\frac{2sinx(1+ctgx)+1}{(1+ ctgx)^2} +\frac{2cos x(1+tgx)-1}{(1+ tgx)^2}$
  
  2. I spot one crucial error. There might be more but you need to correct this one first.
  
  $\displaystyle (\cos^2(x))' = -2\cos(x)\sin(x)$ and $\displaystyle (\sin^2(x))' = 2\sin(x)\cos(x)$
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document contains a math problem involving the calculation of a derivative. The user provides the given function and asks for help in finding the derivative, mentioning that they know the answer is \(-\cos 2x\) but are unable to derive it. The user also shows their work and identifies a potential error in their calculation. Another user points out a crucial error in the derivatives of \(\cos^2(x)\) and \(\sin^2(x)\).
  
  Label: question-and-answer