One of the core concepts of calculus is known as the “derivative”, which is a ratio between two types of changes. For instance, your car’s speedometer gives the you speed in “miles per hour.” This is a derivative—a ratio of the change in distance and the change in time. You can also have a “second derivative,” or a derivative of a derivative. For instance, acceleration is the ratio between the change in speed and the change in time.
However, the notation for the second derivative has always been strange and usually baffles students. In most mathematics textbooks, the notation is presented without explanation for the reasons for why it looks the way it looks. In an effort to provide more background for his students, Jonathan Bartlett decided to pursue the matter further, and figure out why the notation is the way that it is.