Today at school I entered in a problem when the professor asked us to differentiate the following function:
$$f(x)=\arctan\left(\frac {x-1}{x+1}\right)$$
With the basic rules of differentiation I came to a confusing result:
$$f'(x)=\frac 1{1+x^2}$$
And the teacher agreed, and so does Wolfram (I checked at home) but what surprised me is that it's the same derivative as
$$f(x)=\arctan x$$
$$f'(x)=\frac 1{1+x^2}$$
So I'm wondering: is that wrong in some sense ? Are the two function equals indeed ? If I integrate $\frac 1{1+x^2}$ what should I choose from the two ? Are there any other examples of different functions with the same derivative?
Best Answer
Note:
$$\tan(A-B)=\frac{\tan A - \tan B}{1+\tan A \tan B}$$
If $x=\tan A$ and $\tan B=1$, then you get:
$$\tan(A-B)=\frac{x-1}{x+1}$$
So $$\arctan x - B = \arctan\left(\frac{x-1}{x+1}\right)$$ So the functions differ by a constant.
(Well, close enough - they actually differ by a constant locally, wherever both functions are defined. The differences will be constant in $(-\infty,-1)$ and in $(-1,\infty)$, but not necessarily the entire real line.)