# If $y=|\cos x|+|\sin x|,\;find\; \Large\frac{dy}{dx}\; \normalsize at\;x=\Large\frac{2\pi}{3}$

Toolbox:
• before differentiating eliminate modulus by its definition.
$| cos\: x|=cosx\: when \: x \in \bigg[ 0, \large\frac{\pi}{2} \bigg]$
$| sin\: x|=sinx\: when \: x \in [ 0, \pi ]$
$| cos\: x|=-cosx\: when \: x \in \bigg[\large \frac{\pi}{2},\large \frac{3\pi}{2} \bigg]$
$| sin\: x|=-sinx\: when \: x \in [ \pi, 2\pi ]$

$\large\frac{2\pi}{3} \in \bigg[\large \frac{\pi}{2}, \pi \bigg]$
$\Rightarrow y= -cosx+sinx$
$\large\frac{dy}{dx}=sinx+cosx$

$= \large\frac{\sqrt 3}{2}-\large\frac{1}{2}=\large\frac{\sqrt 3-1}{2}\: at \: x=\large\frac{2\pi}{3}$

edited Mar 25, 2013
The text passage contains a mathematical question involving differentiation and provides a detailed solution to the problem.
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
# If $y=|\cos x|+|\sin x|,\;find\; \Large\frac{dy}{dx}\; \normalsize at\;x=\Large\frac{2\pi}{3}$

Toolbox:
• before differentiating eliminate modulus by its definition.
$| cos\: x|=cosx\: when \: x \in \bigg[ 0, \large\frac{\pi}{2} \bigg]$
$| sin\: x|=sinx\: when \: x \in [ 0, \pi ]$
$| cos\: x|=-cosx\: when \: x \in \bigg[\large \frac{\pi}{2},\large \frac{3\pi}{2} \bigg]$
$| sin\: x|=-sinx\: when \: x \in [ \pi, 2\pi ]$

$\large\frac{2\pi}{3} \in \bigg[\large \frac{\pi}{2}, \pi \bigg]$
$\Rightarrow y= -cosx+sinx$
$\large\frac{dy}{dx}=sinx+cosx$

$= \large\frac{\sqrt 3}{2}-\large\frac{1}{2}=\large\frac{\sqrt 3-1}{2}\: at \: x=\large\frac{2\pi}{3}$

edited Mar 25, 2013
user
The text passage contains a mathematical question involving differentiation and provides a detailed solution to the problem.

TYPE: SINGLE_QUESTION_WITH_ANSWER