Sketching Derivative Graphs in A-Level Maths | A-Level数学：导函数图像的绘制技巧

📚 Sketching Derivative Graphs in A-Level Maths | A-Level数学：导函数图像的绘制技巧

Sketching the graph of a derivative function is a core skill in A-Level calculus. It tests not only your algebraic ability to differentiate, but more importantly your conceptual understanding of how a function’s gradient behaves across its domain. This article will guide you through the essential techniques, step by step.

绘制导函数图像是A-Level微积分中的核心技能。它不仅考查你的代数求导能力，更考验你对函数斜率在整个定义域内变化规律的概念性理解。本文将带你逐步掌握这些关键技巧。

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1. The Fundamental Link: f(x) and f'(x) | 基本联系：f(x) 与 f'(x)

Before attempting to sketch f'(x), you must recall the fundamental definition: f'(x) represents the gradient or slope of the tangent line to the curve y = f(x) at any point x. Therefore, when sketching f'(x), you are not drawing the original curve itself, but rather a graph that tells you how steep the original curve is at each x-coordinate.

在尝试绘制 f'(x) 之前，必须回顾其基本定义：f'(x) 表示曲线 y = f(x) 在任意点 x 处切线的斜率。因此，绘制 f'(x) 时，你画的并不是原曲线本身，而是一张反映原曲线在每个 x 坐标上陡峭程度的图。

If f(x) is rising, then f'(x) must be positive; if f(x) is falling, then f'(x) must be negative. Where f(x) has a horizontal tangent (a stationary point), f'(x) must cross the x-axis, meaning f'(x) = 0 at that point.

如果 f(x) 上升，则 f'(x) 必为正；如果 f(x) 下降，则 f'(x) 必为负。当 f(x) 有水平切线（即驻点）时，f'(x) 必与 x 轴相交，即该点处 f'(x) = 0。

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2. Key Features to Identify from f(x) | 从 f(x) 中识别关键特征

To sketch f'(x) accurately, you must first analyse the original function and mark out several crucial features. These features act as anchoring points for your derivative sketch.

要准确绘制 f'(x)，必须先分析原函数并标出几个关键特征。这些特征将作为你绘制导函数图像时的锚点。

• Stationary points: Where f'(x) = 0. These are points where the tangent is horizontal — local maxima, local minima, or stationary inflection points.

• Regions of increase and decrease: Determine on which intervals f(x) is increasing (f'(x) > 0) and on which it is decreasing (f'(x) < 0).

• Vertical asymptotes: If f(x) has a vertical asymptote, f'(x) will typically also tend to infinity (or negative infinity) near that asymptote.

• 驻点：即 f'(x) = 0 的点，这些点的切线水平——包括局部极大值、局部极小值或水平拐点。

• 递增与递减区间：判断 f(x) 在哪些区间递增（f'(x) > 0），在哪些区间递减（f'(x) < 0）。

• 垂直渐近线：若 f(x) 有垂直渐近线，则 f'(x) 在该渐近线附近通常也趋向无穷大（或负无穷大）。

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3. Sign Diagrams: Your First Step | 符号图：你的第一步

A sign diagram for f'(x) is an indispensable tool. It is a number line on which you record where f'(x) is positive, negative, or zero. Constructing this diagram before sketching saves time and prevents sign errors.

f'(x) 的符号图是必不可少的工具。它是一条数轴，在上面标出 f'(x) 为正、为负或为零的区间。在绘制之前先构造符号图，可以节省时间并避免符号错误。

Begin by finding the critical points — where f'(x) = 0 or where f'(x) is undefined. Place these on the number line. Then test a point in each interval to determine the sign of f'(x). Remember: a positive sign means the original curve is rising; a negative sign means it is falling.

首先求出临界点——即 f'(x) = 0 或 f'(x) 无定义的点。将它们标在数轴上，然后在每个区间内选一个测试点来确定 f'(x) 的符号。记住：正号表示原曲线上升；负号表示原曲线下降。

This sign diagram directly translates to the graph of f'(x): wherever the sign is positive, the graph of f'(x) lies above the x-axis; wherever it is negative, it lies below.

这个符号图可以直接转化为 f'(x) 的图像：凡符号为正之处，f'(x) 的图像位于 x 轴上方；凡符号为负之处，则位于 x 轴下方。

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4. Relating Shape to Gradient: Steepness | 将形状与斜率关联：陡峭程度

The height of the graph of f'(x) at a given x tells you how steep f(x) is at that point. If f(x) rises very rapidly, f'(x) will take a large positive value. If f(x) falls steeply, f'(x) will be a large negative value.

f'(x) 图像在某个 x 处的高度告诉你 f(x) 在该点的陡峭程度。如果 f(x) 上升非常快，f'(x) 将取较大的正值。如果 f(x) 急剧下降，f'(x) 将是较大的负值。

Conversely, if f(x) is nearly flat — even if it is not at a stationary point — then f'(x) will be close to zero. This means you should always ask yourself: “How steep is the original curve here?” and translate that answer into the vertical coordinate of f'(x).

反之，若 f(x) 几乎平坦——即使不在驻点——f'(x) 也将接近零。这意味着你应当时刻问自己：”原曲线在这里有多陡？”并将这个答案转化为 f'(x) 的纵坐标。

For example, a cubic curve y = x³ is steep for large |x|, so f'(x) = 3x² grows quadratically — a parabola opening upwards, touching the x-axis at x = 0.

例如，三次曲线 y = x³ 在 |x| 较大时很陡，因此 f'(x) = 3x² 呈二次增长——一条开口向上的抛物线，在 x = 0 处与 x 轴相切。

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5. The Role of Inflection Points | 拐点的作用

An inflection point on f(x) is where the curve changes its concavity — from bending upwards to bending downwards, or vice versa. At such a point, the gradient of f(x) is at a local maximum or minimum value.

f(x) 上的拐点是曲线凹凸性发生改变之处——从凹向上变为凹向下，或相反。在这样的点上，f(x) 的斜率达到局部极大值或极小值。

Consequently, an inflection point on f(x) corresponds to a stationary point of f'(x), but not necessarily a zero of f'(x). In other words, f'(x) has a local maximum or minimum at the x-coordinate of the inflection point.

因此，f(x) 上的拐点对应 f'(x) 的驻点，但不一定是 f'(x) 的零点。换言之，f'(x) 在拐点的 x 坐标处达到局部极大值或极小值。

To locate inflection points, you may set f”(x) = 0. But note: f”(x) = 0 is a necessary but not sufficient condition — you must confirm that the concavity actually changes sign.

要找到拐点，可以令 f”(x) = 0。但注意：f”(x) = 0 是必要但不充分的条件——你必须确认凹凸性确实发生了变号。

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6. Worked Example: A Cubic Function | 实例分析：三次函数

Let us work through a complete example. Consider the function f(x) = x³ − 3x² + 2. This is a cubic with a positive leading coefficient, so it rises on both extremes. We will sketch f'(x) systematically.

我们来完整分析一个例子。考虑函数 f(x) = x³ − 3x² + 2。这是一个首项系数为正的三次函数，因此在两端都上升。我们将系统地绘制 f'(x)。

Step 1 — Differentiate:

第一步——求导：

f'(x) = 3x² − 6x = 3x(x − 2)

Step 2 — Find critical points: Set f'(x) = 0. This gives x = 0 and x = 2. These are the x-coordinates of the stationary points of f(x).

第二步——求临界点：令 f'(x) = 0，解得 x = 0 和 x = 2。它们是 f(x) 驻点的 x 坐标。

Step 3 — Sign analysis: For x < 0, f'(x) is positive; for 0 < x < 2, f'(x) is negative; for x > 2, f'(x) is positive. Thus f(x) rises, then falls, then rises again — confirming that x = 0 is a local maximum and x = 2 is a local minimum.

第三步——符号分析：当 x < 0 时 f'(x) 为正；当 0 < x < 2 时 f'(x) 为负；当 x > 2 时 f'(x) 为正。因此 f(x) 先增、后减、再增——确认 x = 0 是局部极大值，x = 2 是局部极小值。

Step 4 — Sketch f'(x): The derivative is a quadratic with a positive leading coefficient. It is an upward-opening parabola with roots at x = 0 and x = 2. The vertex of this parabola is at x = 1, where f'(1) = −3 — this corresponds to the inflection point of f(x).

第四步——绘制 f'(x)：导函数是首项系数为正的二次函数。它是一条开口向上的抛物线，根在 x = 0 和 x = 2。该抛物线的顶点在 x = 1 处，此时 f'(1) = −3——这对应 f(x) 的拐点。

This example illustrates the full process: differentiate, solve f'(x) = 0, analyse signs, and plot the derivative shape.

这个例子展示了完整的流程：求导、解 f'(x) = 0、分析符号、绘制导函数形状。

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7. Rational Functions and Asymptotes | 有理函数与渐近线

When f(x) is a rational function, additional care is needed near vertical asymptotes and horizontal asymptotes. For a vertical asymptote at x = a, f(x) tends to ±∞, so f'(x) will also tend to ±∞ — the derivative graph will also have a vertical asymptote there.

当 f(x) 是有理函数时，在垂直渐近线和水平渐近线附近需要格外小心。对于 x = a 处的垂直渐近线，f(x) 趋向 ±∞，因此 f'(x) 也趋向 ±∞——导函数图像在该处同样有垂直渐近线。

For a horizontal asymptote y = c as x → ±∞, the gradient of f(x) approaches zero, because the curve becomes flatter and flatter. Hence f'(x) → 0. This means f'(x) has a horizontal asymptote at y = 0 (the x-axis) in that direction.

对于 x → ±∞ 时的水平渐近线 y = c，f(x) 的斜率趋近于零，因为曲线变得越来越平坦。因此 f'(x) → 0。这意味着 f'(x) 在该方向上以 y = 0（x 轴）为水平渐近线。

Notice a subtle point: if f(x) approaches its horizontal asymptote from above, then f'(x) is negative; if from below, f'(x) is positive — at least eventually. Always check the direction of approach.

注意一个微妙之处：若 f(x) 从上方接近其水平渐近线，则 f'(x) 为负；若从下方接近，则 f'(x) 为正——至少最终如此。始终要检查接近的方向。

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8. The Second Derivative: A Helper Tool | 二阶导数：辅助工具

The second derivative f”(x) tells you whether f'(x) is increasing or decreasing. When f”(x) > 0, f'(x) is rising; when f”(x) < 0, f'(x) is falling. This information helps you verify the shape of the derivative graph you are sketching.

二阶导数 f”(x) 告诉你 f'(x) 在递增还是递减。当 f”(x) > 0 时，f'(x) 上升；当 f”(x) < 0 时，f'(x) 下降。这些信息有助于你验证所绘导函数图像的形状。

Moreover, f”(x) = 0 at the inflection points of f(x). At these x-values, the graph of f'(x) has horizontal tangents. So you can pinpoint where the derivative graph “turns around” in terms of steepness.

此外，f”(x) = 0 出现在 f(x) 的拐点处。在这些 x 值上，f'(x) 的图像有水平切线。因此你可以精确确定导函数图像在何处”转向”（即陡峭程度变为最大或最小）。

Note that the graph of f'(x) can also have its own inflection points, which occur where f”'(x) = 0. This is rarely tested at A-Level, but be aware of it for extension problems.

注意 f'(x) 的图像也可以有自己的拐点，这发生在 f”'(x) = 0 处。这在A-Level中很少考到，但在拓展题中要注意。

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9. Common Mistakes to Avoid | 常见错误与避坑指南

Many students make predictable errors when sketching derivative graphs. Recognising these pitfalls will immediately improve your accuracy.

许多学生在绘制导函数图像时都会犯一些可预见的错误。识别这些陷阱能立即提高你的准确性。

• Confusing f'(x) with f(x) itself: When asked to sketch f'(x), some students mistakenly sketch a graph similar to f(x). Always ask: “Where is f(x) steepest? Where is it flat?”

• Forgetting that f'(x) = 0 at every stationary point: Multiple stationary points mean multiple x-axis crossings.

• Ignoring asymptotes: If f(x) has a vertical asymptote, f'(x) must also blow up there. If f(x) levels off, f'(x) must approach zero.

• 混淆 f'(x) 与 f(x) 本身：有些学生在被要求画 f'(x) 时，错误地画出了类似于 f(x) 的图形。永远要问：”f(x) 在哪里最陡？在哪里平坦？”

• 忘记在驻点处 f'(x) = 0：多个驻点意味着 f'(x) 多次穿越 x 轴。

• 忽略渐近线：如果 f(x) 有垂直渐近线，f'(x) 也必在该处趋于无穷大。如果 f(x) 趋于平缓，f'(x) 必趋近于零。

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10. Summary: A Step-by-Step Checklist | 总结：分步检查清单

To conclude, here is a structured checklist you can apply to any f(x) you are asked to differentiate graphically.

最后，这里提供一个结构化的检查清单，你可以将其应用于任何需要绘制导函数图形的 f(x)。

Step | 步骤	Action | 操作
1	Differentiate f(x) to obtain f'(x) | 对 f(x) 求导得到 f'(x)
2	Solve f'(x) = 0 to find x-axis crossings | 解 f'(x) = 0 找出与 x 轴的交点
3	Determine the sign of f'(x) in each interval | 判断每个区间内 f'(x) 的符号
4	Identify vertical asymptotes of f(x) | 找出 f(x) 的垂直渐近线
5	Check f”(x) for inflection points and shape | 用 f”(x) 检查拐点和凹凸形状
6	Sketch f'(x) and verify against f(x) | 绘制 f'(x) 并与 f(x) 对照验证

Practise with a variety of functions — polynomial, rational, exponential, and trigonometric — until the connection between f(x) and f'(x) feels intuitive. This skill will serve you well in both pure mathematics and applied contexts.

请用各种函数——多项式、有理函数、指数函数和三角函数——勤加练习，直到 f(x) 与 f'(x) 之间的联系成为直觉。这项技能在纯数学和应用情景中都将使你受益良多。

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