# Derivatives of arctan and arcsin

• Sep 7th 2006, 06:30 AM
TreeMoney
Derivatives of arctan and arcsin
I just can't seem to figure out how to do this. I'm given f(x) = arctan(x) + artan(1/x) and I need to find the derivative of this function. To be honest I'm not even positive I know exactly what arctan means. Is it the same as tan^-1???
John
• Sep 7th 2006, 06:43 AM
ThePerfectHacker
Quote:

Originally Posted by TreeMoney
I just can't seem to figure out how to do this. I'm given f(x) = arctan(x) + artan(1/x) and I need to find the derivative of this function. To be honest I'm not even positive I know exactly what arctan means. Is it the same as tan^-1???
John

Yes,
$\displaystyle \tan^{-1}$ but that does not mean, $\displaystyle \frac{1}{\tan x}$ it means the inverse function (someone was sleeping in his Calculus Class).
---
If,
$\displaystyle y=\tan^{-1}(x)$
Then,
$\displaystyle y'=\frac{1}{1+x^2}$
And,
$\displaystyle y=\tan^{-1}(1/x)$
Use the rule of chains,
$\displaystyle y'=-\frac{1}{x^2}\cdot \frac{1}{1+(1/x)^2}$
Simplify,
$\displaystyle y'=-\frac{1}{1+x^2}$
Dude they are the same!
Thus,
$\displaystyle \frac{1}{1+x^2}-\frac{1}{1+x^2}=0$
Meaning that function is a constant function.
• Sep 7th 2006, 07:31 AM
CaptainBlack
Quote:

Originally Posted by TreeMoney
I just can't seem to figure out how to do this. I'm given f(x) = arctan(x) + artan(1/x) and I need to find the derivative of this function. To be honest I'm not even positive I know exactly what arctan means. Is it the same as tan^-1???
John

$\displaystyle \arctan(x)$ is defined so that if $\displaystyle y=\arctan(x)$,
then $\displaystyle \tan(y)=x$, but as $\displaystyle \tan$ is periodic with
period $\displaystyle \pi$, it is usual to specify which interval of
length $\displaystyle \pi$ of the real line you want $\displaystyle \arctan$ to be defined on.

Popular choices are $\displaystyle (-\pi/2,\pi/2)$, and $\displaystyle (0,\pi)$

RonL
• Sep 7th 2006, 07:36 AM
ThePerfectHacker
Quote:

Originally Posted by CaptainBlack

$\displaystyle (0,\pi)$

You should not, it is not "well-behaved".
• Sep 7th 2006, 08:55 AM
CaptainBlack
Quote:

Originally Posted by ThePerfectHacker
You should not, it is not "well-behaved".

Just because it is undefined at $\displaystyle \pi/2$ in that range - a bit
severe, but I certainly should have included an end point in that case.
I may concede that I should perhaps have said $\displaystyle [0,\pi)\setminus \{\pi/2 \}$.

RonL
• Sep 7th 2006, 10:06 AM
topsquark
Quote:

Originally Posted by ThePerfectHacker
Yes,
$\displaystyle \tan^{-1}$ but that does not mean, $\displaystyle \frac{1}{\tan x}$ it means the inverse function (someone was sleeping in his Calculus Class).

That's a bit severe. I've never seen "arctan" in any of my Math classes. I only know of it because I learned Calculus on my own from a 20 year old (at the time) textbook. However, so saying, I prefer the notation "atn" better than $\displaystyle tan^{-1}$ because of the possible confusion.

-Dan
• Sep 7th 2006, 04:21 PM
ThePerfectHacker
Quote:

Originally Posted by topsquark
because of the possible confusion.

What confusion? Reciprocal you mean probably. No one ever write a reciprocal function like that.
• Sep 8th 2006, 08:41 AM
TreeMoney
Thanks everyone for you input. This question is actually for my calc 2 class. I have already taken calc 2 but apparently I took the wrong section and now I must take the section which is specifically for the math and sciences and I went through an entire semester of calc 2 and have never seen arctan. I got an A in that class too, no sleepin. :o) Thanks again!! John
• Sep 8th 2006, 08:59 AM
ThePerfectHacker
Quote:

Originally Posted by TreeMoney
I have already taken calc 2 but apparently I took the wrong section and now I must take the section which is specifically for the math and sciences

My College has that too, it offers a more advanced sourse for engineers/physicists and a less advanced course like in your case for i.e. biologists.
• Sep 11th 2006, 06:50 AM
TreeMoney
Quick question.... Is there some sort of program or driver which you guys have installed on your computer which allows you to post these equations the way you do? Reason I ask is because it is rather difficult for me to follow your work when I am trying to figure out how you have done the problem. it says like [tex] blah blah blah. Which is why I feel like my computer is looking for a driver which I don't have installed. Let me in on the know!!! Because you guys are more help then the TA's that I'm supposed to be going to for help. Thanks again. John
• Sep 11th 2006, 07:50 AM
TD!
Quote:

Originally Posted by TreeMoney
Quick question.... Is there some sort of program or driver which you guys have installed on your computer which allows you to post these equations the way you do? Reason I ask is because it is rather difficult for me to follow your work when I am trying to figure out how you have done the problem. it says like [tex] blah blah blah. Which is why I feel like my computer is looking for a driver which I don't have installed. Let me in on the know!!! Because you guys are more help then the TA's that I'm supposed to be going to for help. Thanks again. John

This is a technical problem of the forum, I assume. You don't have to install anything, normally the code in those math-tags generates an image (gif or png) with the equation in a nice graphical style. I believe this "broke" during the upgrade.