The derivative is a function
The lower panel plots a second curve: at each $x$, its height is the slope of $f$ at that same $x$. Where $f$ has a smooth peak — a crest — what is the value of this slope function there?
On $\sin x$, the lower marker traces the cosine wave: at $x = 0$ the tangent climbs at slope $1$ and the marker sits at height $1$; at the crest near $x = 1.57$ the tangent goes flat and the marker crosses zero. Press x²/4 and drag to $x = 3.00$: the f′(x) readout shows $1.500$ while f(x) shows $2.250$ — two different curves, reporting two different things.
Lesson 1 produced one number: the slope at the one point you chose. But nothing in that recipe cares which point. Run the limit at every $x$ and collect the answers, and you have built a new function $f'$, the derivative of $f$ — a slope function whose height at each input is the steepness of $f$ there.
Its most useful reading is the sign, drawn as the dotted fill in the lower panel. Where $f'(x) > 0$ the tangent tilts up, so $f$ is increasing; where $f'(x) < 0$, $f$ is decreasing; where $f'$ crosses zero, $f$ levels off. On the cubic $x^3/6 - x$, the slope function $x^2/2 - 1$ dips below zero between $-1.41$ and $1.41$: $f$ falls over exactly that stretch, cresting at the left end and bottoming out at the right.
On the cubic, the dotted fill below the axis marks where $f'(x) = x^2/2 - 1$ is negative, between $-1.41$ and $1.41$. What is $f$ doing over that stretch?
For $f(x) = x^2/4$ the lower panel is the straight line $f'(x) = x/2$. What does the f′(x) readout show at $x = 3.00$?
Transfer: a function $g$ has $g'(x) = 3$ for every $x$. What must the graph of $g$ look like?
The graph of $f'$ is a complete slope report on $f$: its sign says whether $f$ rises or falls, its size says how steeply, and its zeros mark the horizontal tangents. The next two lessons put that report to work.