Does a line have a tangent line? It would be the line itself. The slope of the tangent line is the slope of the line itself. This is verified by a derivative example. But, isn't the definition of a tangent line "a line that touches at one point"? Doesn't the tangent line to a line touch at infinite points?
$f(x) = 2x+1$
$f'(x) = 2$
This implies a line does have a tangent, right?
Best Answer
It's not so simple to define what is a tangent straight line to a curve. Your definition ''the line that touch at one point'' can work only if the curve is a conic section ( so it is represented by a second degree equation), but also for a simple curve as $y=x^3$ you can see that all the straight lines that are intuitively tangent at some poin, meet the curve at another point, and the only line that ''touch at one point'' is the line $y=0$ that really is a tangent, but through the curve just at the point of tangency.
And you can also easily find functions that has the same line as an ''intuitive'' tangent at more than one point (as a simple exemple: $y=\sin x$ with the lines $y=\pm1$).
So a good definition of tangent can be done only at the some time as the definition of derivative, using a limit. In words:
Developing this intuition we come to the classical definition of the slope of the tangent as: $$ m=\lim_{h\rightarrow 0}\dfrac{f(x+h)-f(x)}{h}=f'(x) $$
Now, if the curve is a stright line, this definition can also be used and, yes, the tangent to a stright line is the same stright line.