when two functions have the same derivative

The derivative of arctan((x-1)/(x+1)) and the derivative of arctan(x) are both equal to 1/(x^2+1). However, do these two functions simply differ by a constant?
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[GreenMeansGOF]

The thing is that



if x<-1 and



If x>-1 so I suppose that it is just a matter of algebraic manipulation. However, the discontinuity may cause problems.

[bashar]

if you assume that ( x = tan u ) the right side will be tan^-1 (tan (u - pi/4)) = u - pi/4 = tan ^ -1 (x) - pi/4 .

and of course, we should note that x doesn't equal -1.

[SuperYoonHo]

thanks so much for your time making these videos

[youngmathematician]

If two functions have the same derivative, then these must be off by a real constant. Let's claim that is the case and that arctan((x-1)/(x+1))-arctan(x)=C, for some real C.

Then,
(To be clear, I first used the trig identity tan(a-b)=(tan(a)-tan(b))/(1+tan(a)tan(b)), hence making the arctans cancel. I then did some algebraic simplifications and cancellations.)

This hence proves that the two functions are off by a constant, and thus have the same derivative. QED

[yoyoezzijr]

They are off by a constant.
Sum formula for inverse tangent: arctan(a) - arctan(b) = arctan(a-b/1+ab)
Thus arctan(x) - arctan(1) = arctan(x-1/1+x) but also arctan(x) - arctan(1) = arctan(x) - π/4
Therefore, arctan(x-1/x+1) = arctan(x) - π/4.

[xinpingdonohoe]

Then put in 0 into both. tan^-1(0)=0, tan^-1((0-1)/(0+1))=-π/4.
Therefore, This question would be a cool way to ask lim(x→-1) tan^-1((x-1)/(x+1)) because you know it's equal to tan^-1(-1)-π/4=-π/2.

[balakumarank]

It's a trigonometric inverse identity
arctan (x) -arctan (y) = arctan (x-y÷(1+xy)

[tbg-brawlstars]

Also, derivative of [-1/(x+1)] and [x/(x+1)] is same!

[rslitman]

Does anyone have the link to the video showing how to use complex numbers to do the integral? Thank you in advance.

[pdwag]

The second integral doesn't work because the domain isn't continuous?

[jaroslavtavgen]

In the function "d(tan(x)^(-1))/dx" x can't be pi and pi/2.

[ДенисКосько-ни]

y=x and y=x+1 have the same derivative!

[wuyqrbt]

Teacher...
I'm sure, alry you have a lot of markers (I'm talking about those ones you used them before). If I was on your shoes, I made something interesting (art) with them (for example *BPRP* sign)

[wuyqrbt]

*tanx* function!? .... COOL .... I love it.
Thank you Teacher ❤️
And Nice Question! (I should think about it)
((Teacher, Please leave the link of videos, you're talking about it on the description ))

[lih]

Plotting it on a graph, arctan((x-1)/(x+1)) is pi/4 less than arctan to the right of -1 and 3pi/4 greater than arctan to the left of -1, with a jump at -1.
Other than that, the curve is exactly the same, what a weird property this is