The change in position – which we’ve called the derivative $s’$ 1 – is what causes a path to turn and circle, to rise and fall. When walking over a hill, the steepness – or we could call it the slope[1,2] – at any point therefore is the derivative $s’$ at that point.
This was obvious.
Something less obvious would be to think of ‘steepness’ and ‘slope’ also for other properties than position.
- What about velocity? It can change,2 so by saying that it has a “slope” we are simply referring to how steep the velocity graph is at a point. We are referring to its derivative $v’$.
- And when saying ‘we’ve experienced steep growth during 2020’ during a financial status meeting in a company, the derivative of the amount of sales3 (so, the change in sales) increased extra much during that year; the slope of the sales curve was extra steep that year.
All in all, slope is a term we might use broadly for any property that we can imagine drawn as a graph.
- ‘Slope of a Line’ (web page), Math Open Reference, 2011, www.mathopenref.com/coordslope.html (accessed Jun. 14th, 2020)
- ‘Slope and y-intercept’ (web page), math.com, 2005, www.math.com/school/subject2/lessons/S2U4L2GL.html (accessed Jun. 14th, 2020)