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 $\displaystyle (\sin^2(x))' = 2\sin(x)\cos(x)$
The text passage contains a mathematical question asking for the derivative of a given function, and the correct answer is provided. Additionally, there is a correction to the derivative formulas for \(\cos^2(x)\) and \(\sin^2(x)\).
single_question_with_answer
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
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 $\displaystyle (\sin^2(x))' = 2\sin(x)\cos(x)$
user
The text passage contains a mathematical question asking for the derivative of a given function, and the correct answer is provided. Additionally, there is a correction to the derivative formulas for \(\cos^2(x)\) and \(\sin^2(x)\).

TYPE: SINGLE_QUESTION_WITH_ANSWER