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 **Δ***T*/Δ*x*.

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 Δ*T* is -3.1. We move 5.5 spaces on the x-axis so that Δ*x* 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.)