Ask a Question
  
  
  
Top Posters
Since Sunday
7
5
4
4
4
4
4
4
4
4
4
4
New Topic  
wrote...
Administrator
Educator
Posts: 23920
5 months ago
Derive:

\(y=\operatorname{arccot}x+\arctan \left(\frac{2+x}{1-2x}\right)\)

Before we start, recall:

\(\frac{d}{dx}\operatorname{arccot}x=\frac{-1}{1+x^2}\)
\(\frac{d}{dx}\arctan x=\frac{1}{1+x^2}\)

Now, let's return to the problem, the left side is easy, it becomes: \(\frac{dy}{dx}\), but the right side, specifically the tangent part requires the chain rule:

\(\frac{dy}{dx}=\frac{-1}{1+x^2}+\frac{1}{1+\left(\frac{2+x}{1-2x}\right)^2}\left(\frac{\left(2+x\right)\left(-2\right)-\left(1-2x\right)}{\left(1-2x\right)^2}\right)\)

The derivative is done. Now we clean it:

\(\frac{dy}{dx}=\frac{-1}{1+x^2}+\frac{1}{1+\left(\frac{2+x}{1-2x}\right)^2}\left(\frac{-4-2x-1+2x}{\left(1-2x\right)^2}\right)\)

Clean more:

\(\frac{dy}{dx}=\frac{-1}{1+x^2}+\frac{1}{1+\left(\frac{2+x}{1-2x}\right)^2}\left(\frac{-5}{\left(1-2x\right)^2}\right)\)

Clean more:

\(\frac{dy}{dx}=\frac{-1}{1+x^2}+\frac{-5}{\left[1+\left(\frac{2+x}{1-2x}\right)^2\right]\left(1-2x\right)^2}\)

If you want, you can combine more, but this is fine.
Read 166 times
The best way to say thank you is with a positive review:

  https://trustpilot.com/review/biology-forums.com 

Your support goes a long way!


Make a note request here
Related Topics
New Topic      
Hold tight!  A Bio Forums Expert has been contacted to start answering this thread.
Explore
Post your homework questions and get free online help from our incredible volunteers.
Learn More
Improve Grades
Help Others
Save Time
Accessible 24/7
  155 People Browsing
 124 Signed Up Today
Related Images
 5443
 783
 53