Derivatives of arctan and arcsin

**TreeMoney** · September 7th 2006, 06:30 AM

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???
Thanks in advance
John

**ThePerfectHacker** · September 7th 2006, 06:43 AM

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???
Thanks in advance
John

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

**CaptainBlack** · September 7th 2006, 07:31 AM

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???
Thanks in advance
John

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

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

RonL

**ThePerfectHacker** · September 7th 2006, 07:36 AM

Originally Posted by CaptainBlack

$(0,\pi)$

You should not, it is not "well-behaved".

**CaptainBlack** · September 7th 2006, 08:55 AM

Originally Posted by ThePerfectHacker

You should not, it is not "well-behaved".

Just because it is undefined at $\pi/2$ 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 $[0,\pi)\setminus \{\pi/2 \}$ .

RonL

**topsquark** · September 7th 2006, 10:06 AM

Originally Posted by ThePerfectHacker

Yes,
$\tan^{-1}$ but that does not mean, $\frac{1}{\tan x}$ 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 $tan^{-1}$ because of the possible confusion.

-Dan

**ThePerfectHacker** · September 7th 2006, 04:21 PM

Originally Posted by topsquark

because of the possible confusion.

What confusion? Reciprocal you mean probably. No one ever write a reciprocal function like that.

**TreeMoney** · September 8th 2006, 08:41 AM

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.

) Thanks again!! John

**ThePerfectHacker** · September 8th 2006, 08:59 AM

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.

**TreeMoney** · September 11th 2006, 06:50 AM

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

**TD!** · September 11th 2006, 07:50 AM

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.

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