Not the slope of a graph
Most courses introduce the derivative as the slope of a tangent line, which puts a picture before the idea. The graph is a consequence, not the definition. Before any graph exists, a function is a machine: push the input a hair, and the output moves by some amount. Compare those two movements and you have the derivative.
That comparison is the ratio from chapter one, doing its one job. The input nudge is called 1;
whatever the output nudge is next to it, that number is df/dx. Nothing is being
divided in the arithmetic sense, and nothing is approaching anything yet.
Which dissolves the usual sticking point. A nudge is too small to see — so how can you compare two invisible things? The same way we compared two fruits without ever learning their gram counts. A comparison index doesn't need absolute sizes. We never find out how big the nudge is, and we never need to.
Then the economics falls out for free. Marginal utility, marginal cost, marginal revenue: each is "one more unit of input, how much more output?" That's a nudge ratio with a different word on it.
Chapters in the video
- 0:00Isn't a derivative just the slope?
- 0:39What the dictionary actually says
- 1:53What a function really is
- 2:37Nudging the input point
- 3:13dx, df, and df/dx under the microscope
- 4:24Where a graph comes from, and the derivative f′
- 7:48Marginal utility is just a derivative
Read it instead
The written version, aimed at the economics side of it: "marginal" is just the economist's word for a derivative — including why MR = MC is the first-derivative test in disguise.
Walk inside it
Nudges are easier to believe when you can stand on the surface they live on. Drop a function into Open World Discover and the island becomes that graph.