The form I learned as an undergraduate was the one that says that (under appropriate conditions) for any points a and b, there is a point c such that the derivative at c is equal to the slope of the secant determined by a and b. (This of course makes it easy to prove the zero-derivative theorem.) However, in googling for the Intermediate Value Theorem for Derivatives, I find the more literal one that says that if w is between the derivative at a and the derivative at b, then there is a point c such that the the derivative of c is w.
So, are there in fact two widely-accepted forms of the Intermediate Value Theorem for Derivatives? – and if so, are they equivalent? Of course, they would be trivially equivalent in that they are both true (because they have the same truth-value, which is the truth-table definition of equivalence), but I mean, is one easily manipulated into the other?