The normal-line triangle transform

For a point A = (x, f(x)) on the curve: drop straight down to B = (x, 0), then draw the line normal to the curve at A and find where it crosses the x-axis at C. Triangle ABC's signed area becomes y′(x) — plotted here on the same axes as f(x) itself.

y′(x)  =  ½ f′(x) [f(x)]²
f(x) y′(x)