Velocity Is the Slope of Position, Acceleration the Slope of Velocity
A single particle moves along a line. Drag time and it glides from place to place. But look closely: over some stretches it races through, over others it nearly stands still. Even in the same one second it covers different ground. How fast the position changes like this is exactly the velocity. And even that velocity changes from moment to moment, speeding up and then slowing again. How fast the velocity changes is the acceleration. Going from position to velocity, and from velocity to acceleration, the thing you do at each step up is the same: you ask how fast it is changing. That is differentiation. This one simple operation holds up the whole of particle kinematics, and the reverse, stacking things back up by integration, retraces the way down to position again.
First, drag time t to move the particle. As you push the slider, the particle slides along the line, and its position x at each instant shows below. It starts off slow, sweeps through the middle, and slows again near the end. The markers, dropped at equal time intervals, make it plain at a glance: where they are crowded it moved slowly, where they are spread apart it moved fast. For now, hold on to just one thing: fix the time, and the position is fixed. This relation x(t) holds the entire motion. From the next scene on, we dig into how fast that position changes.
Now spread the position out as a graph against time. The horizontal axis is time t, the vertical axis is position x. Drag the point along the curve with time, and the tangent line touching it moves too. The slope of that tangent is the velocity. v = dx/dt: how much the position changes when just a sliver of time passes. Where the curve is steep, in the middle, the slope is large, so it is fast; near the flat ends the slope is close to zero, so it is essentially stopped. Velocity is not the position itself but its rate of change, not the height but the steepness of the hill. That is why something can be far away yet momentarily at rest, and right near the start yet whipping along.
Do the very same thing, one floor up. This time it is velocity graphed against time. Except that the vertical axis now reads velocity v, it looks just like the last scene. Drag the point with time, and the slope of the tangent is now the acceleration. a = dv/dt: how fast the velocity is changing. Just after the start the velocity keeps growing, so the slope is positive, that is speeding up. At the instant velocity reaches its peak the graph is a crest, so the slope is zero, and in that moment the acceleration is zero too. After that the velocity drops, so the slope is negative, that is slowing down. The point is that the operation is identical: differentiation, taking the rate of change. Apply it to position and you get velocity; apply it to velocity and you get acceleration.
Stack the three graphs and take them in at a glance. The top is position x, the middle is velocity v, the bottom is acceleration a. Switch the motion with the buttons and watch how the shapes lock together. For constant velocity the position is a straight ramp, the velocity is a flat horizontal line, and the acceleration is zero. For constant acceleration the position is a curving parabola, the velocity is a straight rising line, and the acceleration is a steady horizontal line. The rule is always the same: the slope of the upper graph is the height of the one just below it. Where position is steep, velocity is high; where velocity is steep, acceleration is large. One step down is one differentiation, and that chain ties the three rows into one.
If differentiation runs one way, we ought to be able to go back the other way. Return to the velocity graph and drag time t. The region under the curve, from zero up to now, fills in with color. This shaded area is the distance traveled so far, that is, the change in position. Since velocity times time is distance, stacking up each instant's v across time gives the total displacement as an area. This is integration. Where differentiation pulled a slope off the curve, integration gathers the area beneath it. Drag all the way and the summed area matches exactly the distance from start to finish. So even knowing only the acceleration, you can stack it up into velocity, and stack velocity into position. Differentiation and integration are two directions on the same ladder.