上一片学会了“无限靠近”。现在把极限用在速度上：小算盘机器人 🧮 开着小车，仪表盘上那个数字，到底是怎么算出来的？Last leaf we learned to “get infinitely close”. Now we put limits to work on speed: the little abacus robot 🧮 drives a car — how does the dashboard come up with the number on the dial?
第 1 步：平均速度 vs 瞬时速度Step 1: Average speed vs instant speed
先说平均速度——它很好算：总路程 ÷ 总时间。Average speed comes first — and it is easy: total distance ÷ total time.
机器人用 2 小时开了 120 公里，平均速度就是 120÷2＝60 公里/时。但它一路上有时快、有时慢，平均值把这些起伏都抹平了。你在“某一刻”的速度，叫瞬时速度：平均回答“这段路整体多快”，瞬时回答“此刻有多快”。The robot drives 120 km in 2 hours, so the average speed is 120 ÷ 2 = 60 km/h. But along the way it sped up and slowed down — the average smooths all that away. The speed at one particular moment is the instant speed: average tells you how fast a whole trip was, instant tells you how fast right now.
那“此刻”这么短的一瞬间，怎么量？So how do you measure speed during one instant?
第 2 步：把时间间隔压到无限小Step 2: Squeeze the time gap toward zero
还是用极限。想量某一刻的速度，就先量一小段时间里的平均速度；然后把这段时间压小：1 分钟、1 秒、0.001 秒……平均速度会越来越稳，奔向一个值。这个极限值就是那一刻的瞬时速度——速度表上的 60，本质上是一个极限。Use a limit, again. To measure the speed at one moment, first measure the average over a tiny time gap; then squeeze the gap: one minute, one second, 0.001 second… The average settles down and heads toward a single value. That limit is the instant speed at that moment — the “60” on the dial is really a limit.
换个镜头看图：速度其实是“陡不陡”的问题。Now switch to a graph: speed is really about steepness.
第 3 步：导数是切线的斜率Step 3: A derivative is the tangent's slope
把路程画成曲线，平均速度就是两点连线的斜率；让两点越靠越近，连线变成贴在曲线上的切线，它的斜率就是该点的导数，记作 f′(x)。坡越陡，变化越快；切线水平，说明这一刻正好“停了一下”，斜率为 0。Plot the trip as a curve: the average speed is the slope of the line joining two points. Slide the points together and the line becomes the tangent that hugs the curve — its slope is the derivative at that point, written f′(x). A steeper slope means faster change; a flat tangent means the curve pauses for an instant, with slope 0.
🎮 滑动的切线（30 秒）🎮 The sliding tangent (30 seconds)
道理讲完了。拖动滑块让红点在曲线 y=sin x 上滑动，看金色切线怎么跟着转、斜率怎么变；再完成 4 个任务。Theory done. Drag the slider to slide the red ball along the curve y=sin x — watch the golden tangent turn and the slope change; then finish 4 tasks.
一句话记住它：导数 = 曲线在某点的切线斜率 = 那一瞬间的变化快慢。Remember it in one line: a derivative is the slope of the tangent at a point — how fast things change at that instant.
平均速度 = 总路程 ÷ 总时间；瞬时速度回答“此刻有多快”Average speed = total distance ÷ total time; instant speed answers “how fast right now?”把时间间隔压到无限小，平均速度的极限就是瞬时速度Squeeze the time gap toward zero: the limit of average speeds is the instant speed导数就是切线斜率：越陡变化越快，水平表示斜率为 0The derivative is the tangent's slope: steeper means faster change, flat means slope 0
内容参考 OpenStax 等公开教材，多来源核对 · AI 生成、人工审核 · 发现错误欢迎指正，帮这片叶子长得更好。Based on OpenStax and other open textbooks, cross-checked across sources · AI-generated, human-reviewed · Spotted a mistake? Tell us — help this leaf grow.
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