Derivatives & Slope
Walk a curved hillside and the ground tilts differently under every footstep; the height follows a rule as plain as `x**2`, yet its steepness keeps shifting, and pinning that steepness to one exact spot is the whole move here.
- ▸A curve has no single steepness: it tilts differently at every point, and those tilts form a second curve.
- ▸To measure steepness at one point, shrink the gap between two footprints to nothing; the tilt settles on one value.
- ▸The sign of that tilt is a downhill compass, how a model walks its loss curve to the bottom.
The last topic left you standing on a bending curve where the same step to the right lifts the output by different amounts depending on where you stand. That raises an obvious question the curve never answered: how steep is the ground at one exact spot, not averaged over a stretch? Pick a point on the hillside. Draw a straight line to a second point nearby and you get its average tilt between the two, rise over run. Now slide that second point closer. The line pivots, and as the gap closes toward nothing, its tilt stops moving and settles on a single number: the steepness right where you are standing. That settled line grazing the curve at one point is the tangent line, the connecting chord between two points is the secant line, and the number the tangent's tilt settles on is the derivative at that point.