What is the result of deriving a polygon?

Question

If you define a polygon - say, for simplicity, a triangle - as a list of functions, defined piecewise, for example:

• $a(x)=2x$ defined on $[0,1]$
• $b(x)=-2x+2$ defined on $[1,2]$
• $c(x)=0$ defined on $[0,2]$

What do you get by deriving the functions on their intervals and plotting $a'(x)$, $b'(x)$ and $c'(x)$? (In this specific case, the result is made of three segments parallel to the x axis, but what is the result of any given polygon?)

Answer 1

$(ax+b)' = a$, thus if you define a set of line segments as you did, their derivatives will be defined as $y = a_i$ for $x \in [p_i, q_i]$ (here $a$ comes from the line equation and $[p, q]$ is the interval on which it is defined). These lines have a property that $\sum a_i|p_i - q_i| = 0$.