How do you measure how steep a curve is at a single point?

You measure how steep a curve is at one point by drawing the straight line that just touches the curve there and measuring the steepness of that line. That steepness is called the derivative. For a function of one variable, the derivative at a chosen input is the slope of the tangent line to the graph at that point, and it is often described as the instantaneous rate of change.

The picture is the same one you met with a speedometer. Imagine the graph of your position over time. The steepness of the curve at a moment is your speed at that moment: a steep stretch means fast, a flat stretch means you are standing still. Between two moments you can always measure the steepness of the straight line joining them, which is an average. The trick is to slide one of the two points towards the other. The joining line swings round, and its slope settles towards one value. That value is the slope at the single point.

Why does a straight line describe a curve at all? Because the tangent line is the best linear approximation of the function near that input value. Zoom in far enough on a smooth curve and it looks almost straight, and the tangent is that straight line. The process of finding a derivative is called differentiation.

The idea is older than calculus. In the 1630s the French mathematician Pierre de Fermat developed a technique called adequality to calculate tangents and used it on the parabola. Later Leibniz defined the tangent as the line through a pair of infinitely close points on the curve. The derivative is useful well beyond lines: in physics, the first derivative of position with respect to time is velocity, and the second derivative is acceleration.