We wondered what a derivative is. Imagine you have a graph with temperature on the y-axis and x on the x-axis. If you pick two points on the graph you can calculate the difference in their y values and the difference in their x values. Dividing those, you would get ΔTx.

In the top-left graph we pick two points far apart. Going from the first point to the second we move 3.1 spaces down on the y-axis, so that ΔT is -3.1. We move 5.5 spaces on the x-axis so that Δx is 5.5. Doing the math it’s -0.56.

But look, if we pick different points we get different values. In the top-right we get -1.67, and in the bottom-left we get -0.36. It depends on what two points we pick.

Now this is a derivative: what if we say the two points we pick are zero distance apart so essentially they are the same point? That is dT/dx, shown in the bottom-right. Each point on the graph will have a different dT/dx value, which is the derivative at that point.

This is now calculus btw, because we talked about two points zero distance apart. (Or an “infinitesimal distance apart” which means infinitely close together.)