Is this interpolation, does it have a name? I was waching Signle Variable Calculus MIT lectures (I stop on 9 about linear approximation) I was also learn interpolation at my university and I thought that I'll create my own equation for interpolation to better understand it. I've ask this question Function with single formula that have non zero value only in one small range which give me all I need. 
So I came up with this equation:
$ I(x) = \sum_{i=0}^{n-1}\frac{(x-x_i)+|x-x_i|}{2|x-x_i|} * \frac{(x_{i+1}-x)+|x_{i+1}-x|}{2|x_{i+1}-x|} * (f'(x_i)*x+f(x_i)) $
Is this the interpolation that interpolate using tangent lines? Does someone else discovered this type of interpolation, if it's correct, does it have a name?
I've use wolfram alpha to test: http://bit.ly/Waq7qJ instead of tangent lines I just use lines $x+b$.
UPDATE:
I have a function and have list of $x$ and can calculate list of $y$ out of them so I have pairs of $x_i$ and $f(x_i)$ which are numbers. 2 fractions in equations are just cut function that draw only fraction in parantheses, the fraction is $<x_i, x_{i+1}>$ If I add 2 ranges that are next to each other I'll get line without breaks.
The part in parenteses is my line function, $f'(x_i)$ is the tangent of the line (a number) so its $a$ in $a*x+b$ so if I multiply it by $x$ I have a line (a function so it also make $I(x)$ a function). To have the line in correct place I need to shift it so I add $f(x_i)$ so it's in correct place.
 A: Sorry I poorly read your post; it turns out that your idea is correct.
As far as I understand it, you are creating a function which connects the points $(x_i,f(x_i))$ together with straight line segments. A few notes:


*

*I think the linear part is wrong. It should be $$f(x_i)+f'(x_i)\cdot(x-x_i)$$ since you add on the change, equal to rate of change $f'(x_i)$ times change $\Delta x=x-x_i$, to the value $f(x_i)$.

*Your formula returns $0$ outside the range $(x_0,x_n)$. Not that that's a bad thing, just something to note. More importantly, 

*Your formala (as written) fails to exist at the points $x_i$ themselves, because then you are dividing by $0$. I think you should replace expressions of the form $$\frac{a+|a|}{2a}$$ with simply $$\text{sign}(x):=\begin{cases} 1&x>0 \\ 0&x\le 0\end{cases}$$ which returns the same results, and still exists when $a=0$ (also, I just noticed that you are using absolute value signs in the denominator, which I do not think is necessary). However, in this case that does not solve the problem because now the formula would return $0$ at the interpolation "sample" points. The only solution I can think of (for getting an expression in closed form) is to use a limit like so: $$\lim_{z\to a^+}\frac{z+|z|}{2z}$$ 
When $a=0$ this returns $1$ which is what we want.

*As to names, you may want to research "spline interpolation" - this is more or less a simple case of that.
