Derivative Graphs: Sketching and Feature Analysis | 导函数图像：绘制与特征分析

📚 Derivative Graphs: Sketching and Feature Analysis | 导函数图像：绘制与特征分析

The graph of a derivative, often denoted as \( f'(x) \), is one of the most powerful tools in calculus. It encodes the rate of change of a function \( f(x) \) at every point, and once you learn to read it, you can reconstruct the behaviour of the original function without knowing its explicit formula. This article provides a systematic approach to sketching derivative graphs and interpreting their key features.

导函数图像，通常记为 \( f'(x) \)，是微积分中最强大的工具之一。它编码了函数 \( f(x) \) 在每个点的变化率，一旦你学会阅读它，即使不知道原函数的显式表达式，也能重构原函数的行为。本文将系统讲解如何绘制导函数图像以及如何分析其关键特征。

1. What Is a Derivative Graph? | 什么是导函数图像？

A derivative graph plots the slope of the tangent line to \( f(x) \) at each value of \( x \). For a point \( x=a \), the value \( f'(a) \) is the instantaneous rate of change of \( f \) at that point. When \( f'(a) > 0 \), the function is increasing at \( a \); when \( f'(a) < 0 \), it is decreasing; when \( f'(a) = 0 \), the tangent is horizontal.

导函数图像描绘的是 \( f(x) \) 在每个 \( x \) 处切线的斜率。对于点 \( x=a \)，\( f'(a) \) 是 \( f \) 在该点的瞬时变化率。当 \( f'(a) > 0 \) 时，函数在 \( a \) 处递增；当 \( f'(a) < 0 \) 时，函数递减；当 \( f'(a) = 0 \) 时，切线水平。

Derivative = Slope of tangent = \(\frac{dy}{dx}\) | 导数 = 切线斜率 = \(\frac{dy}{dx}\)

It is essential to distinguish between the graph of \( f \) and the graph of \( f’ \). The y-coordinate on \( f’ \) is not the value of \( f \) but the slope of \( f \). This distinction is the starting point for all feature analysis.

必须区分 \( f \) 的图像与 \( f’ \) 的图像。\( f’ \) 图像上的 y 坐标不是 \( f \) 的值，而是 \( f \) 的斜率。这种区分是所有特征分析的起点。

2. Key Connections: \( f \) and \( f’ \) | 关键联系：\( f \) 与 \( f’ \)

The behaviour of \( f’ \) directly reflects the shape of \( f \). The table below summarises the most important relationships.

\( f’ \) 的行为直接反映 \( f \) 的形状。下表总结了最重要的关系。

Feature of \( f’ \)	Feature of \( f \)
\( f'(x) > 0 \)	\( f \) is increasing (slope positive)
\( f'(x) < 0 \)	\( f \) is decreasing (slope negative)
\( f'(x) = 0 \)	\( f \) has a horizontal tangent (stationary point)
\( f’ \) is increasing	\( f \) is convex / concave up
\( f’ \) is decreasing	\( f \) is concave down
\( f’ \) has a local maximum/minimum	\( f \) has an inflection point (where concavity changes)

注意：表中的表述要对应准确。\( f’ \) 递增意味着 \( f \) 的斜率越来越大，因此 \( f \) 凹向上；\( f’ \) 递减则 \( f \) 凹向下。\( f’ \) 的极值点对应 \( f \) 的拐点（凹凸性改变点）。

3. Drawing \( f’ \) from a Given \( f \) | 由已知 \( f \) 绘制 \( f’ \)

To sketch the derivative graph directly from the graph of \( f \), follow these steps:

要从 \( f \) 的图像直接绘制导函数图像，请遵循以下步骤：

Mark all horizontal tangents. At these points, \( f'(x) = 0 \), so the graph of \( f’ \) crosses the x-axis.
Determine the sign of the slope on each interval. If \( f \) is rising, \( f’ \) lies above the x-axis; if falling, below.
Estimate the magnitude of the slope. Steeper sections of \( f \) correspond to larger absolute values of \( f’ \).
Connect the points smoothly, respecting increasing and decreasing behaviour of the slope itself.

标出所有水平切线的位置。在这些点处，\( f'(x) = 0 \)，因此 \( f’ \) 的图像与 x 轴相交。

判断每个区间上斜率的正负。若 \( f \) 上升，\( f’ \) 在 x 轴上方；若下降，则在 x 轴下方。

估计斜率的大小。\( f \) 越陡峭的部分，对应 \( f’ \) 的绝对值越大。

平滑地连接各点，同时注意斜率本身的增减趋势。

Example: For \( f(x) = x^3 – 3x \), we have \( f'(x) = 3x^2 – 3 \). The derivative is zero at \( x = ±1 \).

例如：对于 \( f(x) = x^3 – 3x \)，有 \( f'(x) = 3x^2 – 3 \)。导数在 \( x = ±1 \) 处为零。

4. Drawing \( f \) from a Given \( f’ \) | 由已知 \( f’ \) 绘制 \( f \)

Recovering the shape of \( f \) from \( f’ \) is a matter of integration, but for sketching purposes we use qualitative reasoning. The process is the reverse of the previous section.

从 \( f’ \) 反推 \( f \) 的形状需要通过积分，但在绘图时我们使用定性推理。这一过程与上一节相反。

Identify intervals where \( f’ > 0 \): \( f \) increases there.
Identify intervals where \( f’ < 0 \): \( f \) decreases there.
Where \( f’ \) changes from positive to negative, \( f \) has a local maximum; from negative to positive, a local minimum.
Use the slope of \( f’ \) to determine concavity: if \( f’ \) is increasing, \( f \) is concave up; if decreasing, concave down.

确定 \( f’ > 0 \) 的区间：\( f \) 在这些区间递增。

确定 \( f’ < 0 \) 的区间：\( f \) 在这些区间递减。

当 \( f’ \) 从正变负时，\( f \) 有局部极大值；从负变正时，有局部极小值。

利用 \( f’ \) 的斜率判断凹凸性：\( f’ \) 递增则 \( f \) 凹向上，\( f’ \) 递减则 \( f \) 凹向下。

Note that \( f \) is only determined up to a vertical translation. The graph of \( f \) can be shifted up or down without changing \( f’ \).

注意：\( f \) 只能确定到相差一个垂直平移。\( f \) 的图像上下移动不会改变 \( f’ \)。

5. Stationary Points and Their Classification | 驻点及其分类

A stationary point of \( f \) occurs where \( f'(x) = 0 \). The sign pattern of \( f’ \) around that point determines its nature.

\( f \) 的驻点出现在 \( f'(x) = 0 \) 处。\( f’ \) 在该点附近的符号变化决定了驻点的性质。

Change in \( f’ \)	Type of stationary point
+ → −	Local maximum
− → +	Local minimum
No sign change	Stationary inflection / horizontal inflection

符号变化：+ → − 为局部极大值；− → + 为局部极小值；无符号变化为水平拐点（驻点型拐点）。

For example, \( f(x) = x^3 \) has \( f'(x) = 3x^2 \). At \( x=0 \), \( f'(0)=0 \), but \( f’ \) is positive on both sides, so the point is a stationary inflection, not an extremum.

例如，\( f(x) = x^3 \) 中 \( f'(x) = 3x^2 \)。在 \( x=0 \) 处 \( f'(0)=0 \)，但 \( f’ \) 在两侧均为正，因此该点是水平拐点而非极值点。

6. Concavity and Inflection Points | 凹凸性与拐点

Concavity is determined by the sign of the second derivative \( f”(x) \), but it can also be read from the graph of \( f’ \). If \( f’ \) is increasing, then \( f”(x) > 0 \) and \( f \) is concave up (cup-shaped). If \( f’ \) is decreasing, then \( f”(x) < 0 \) and \( f \) is concave down (cap-shaped).

凹凸性由二阶导数 \( f”(x) \) 的符号决定，但也可以从 \( f’ \) 的图像中读出。若 \( f’ \) 递增，则 \( f”(x) > 0 \)，\( f \) 凹向上（杯形）。若 \( f’ \) 递减，则 \( f”(x) < 0 \)，\( f \) 凹向下（帽形）。

An inflection point on \( f \) occurs where the concavity changes, which corresponds to a local extremum on the graph of \( f’ \). At such a point, \( f’ \) has a turning point, and \( f”(x) \) changes sign.

\( f \) 的拐点出现在凹凸性改变的地方，对应 \( f’ \) 图像上的局部极值点。在该点，\( f’ \) 有转折，且 \( f”(x) \) 改变符号。

Inflection point: \( f”(x) = 0 \) and \( f” \) changes sign | 拐点：\( f”(x) = 0 \) 且 \( f” \) 变号

When sketching \( f’ \), look for points where the graph of \( f’ \) reaches a maximum or minimum; these mark inflection points on \( f \).

绘制 \( f’ \) 时，要寻找 \( f’ \) 图像达到最大值或最小值的点；这些点标记了 \( f \) 上的拐点。

7. Asymptotes and End Behaviour | 渐近线与端部行为

The end behaviour of \( f \) affects the end behaviour of \( f’ \). If \( f \) approaches a finite horizontal asymptote, then \( f'(x) \to 0 \) as \( x \to ±∞ \). If \( f \) grows linearly, \( f’ \) tends to a constant. If \( f \) grows quadratically, \( f’ \) grows linearly, and so on.

\( f \) 的端部行为会影响 \( f’ \) 的端部行为。若 \( f \) 趋近于有限水平渐近线，则 \( x \to ±∞ \) 时 \( f'(x) \to 0 \)。若 \( f \) 线性增长，\( f’ \) 趋于常数；若 \( f \) 二次增长，\( f’ \) 线性增长，依此类推。

Vertical asymptotes of \( f \) often produce vertical asymptotes or infinite discontinuities in \( f’ \). For example, \( f(x) = \frac{1}{x} \) has \( f'(x) = -\frac{1}{x^2} \), and both have a vertical asymptote at \( x=0 \).

\( f \) 的垂直渐近线通常导致 \( f’ \) 出现垂直渐近线或无穷间断。例如 \( f(x) = \frac{1}{x} \) 的导数为 \( f'(x) = -\frac{1}{x^2} \)，两者均在 \( x=0 \) 处有垂直渐近线。

When sketching \( f’ \), always consider the limits as \( x \to ±∞ \). This gives the overall trend and helps place the graph correctly.

绘制 \( f’ \) 时，始终考虑 \( x \to ±∞ \) 的极限。这给出了整体趋势，有助于正确放置图像。

8. Worked Example 1: Polynomial \( f(x) = x^3 – 6x^2 + 9x \) | 实例一：多项式 \( f(x) = x^3 – 6x^2 + 9x \)

Let us sketch \( f \) and its derivative \( f'(x) = 3x^2 – 12x + 9 \). First factor the derivative.

让我们绘制 \( f \) 及其导数 \( f'(x) = 3x^2 – 12x + 9 \) 的图像。首先对导数因式分解。

\( f'(x) = 3(x^2 – 4x + 3) = 3(x-1)(x-3) \)

The derivative is zero at \( x=1 \) and \( x=3 \). Using the sign of \( f’ \):

导数在 \( x=1 \) 和 \( x=3 \) 处为零。根据 \( f’ \) 的符号：

For \( x < 1 \), \( f'(x) > 0 \): \( f \) is increasing.
For \( 1 < x < 3 \), \( f'(x) < 0 \): \( f \) is decreasing.
For \( x > 3 \), \( f'(x) > 0 \): \( f \) is increasing again.

当 \( x < 1 \) 时，\( f'(x) > 0 \)：\( f \) 递增。

当 \( 1 < x < 3 \) 时，\( f'(x) < 0 \)：\( f \) 递减。

当 \( x > 3 \) 时，\( f'(x) > 0 \)：\( f \) 再次递增。

Hence \( x=1 \) gives a local maximum and \( x=3 \) gives a local minimum. The second derivative \( f”(x) = 6x – 12 \) is zero at \( x=2 \), so \( f \) has an inflection point there. The graph of \( f’ \) is an upward-opening parabola with vertex at \( x=2 \), which matches the inflection point of \( f \).

因此 \( x=1 \) 处为局部极大值，\( x=3 \) 处为局部极小值。二阶导数 \( f”(x) = 6x – 12 \) 在 \( x=2 \) 处为零，所以 \( f \) 在此有拐点。\( f’ \) 的图像是开口向上的抛物线，顶点在 \( x=2 \)，与 \( f \) 的拐点对应。

9. Worked Example 2: Trigonometric \( f(x) = \sin x \) | 实例二：三角函数 \( f(x) = \sin x \)

The derivative of \( \sin x \) is \( \cos x \). The graph of \( \cos x \) crosses the x-axis at \( x = ±\frac{\pi}{2}, ±\frac{3\pi}{2}, \ldots \), which are exactly the points where \( \sin x \) has horizontal tangents.

\( \sin x \) 的导数是 \( \cos x \)。\( \cos x \) 的图像在 \( x = ±\frac{\pi}{2}, ±\frac{3\pi}{2}, \ldots \) 处与 x 轴相交，这些正是 \( \sin x \) 有水平切线的点。

When \( \sin x \) is increasing (from \( -\frac{\pi}{2} \) to \( \frac{\pi}{2} \)), \( \cos x \) is positive. When \( \sin x \) is decreasing, \( \cos x \) is negative. The maximum of \( \cos x \) at \( x=0 \) corresponds to the steepest positive slope of \( \sin x \), which occurs at \( x=0 \).

当 \( \sin x \) 递增时（从 \( -\frac{\pi}{2} \) 到 \( \frac{\pi}{2} \)），\( \cos x \) 为正。当 \( \sin x \) 递减时，\( \cos x \) 为负。\( \cos x \) 在 \( x=0 \) 处的最大值对应 \( \sin x \) 在 \( x=0 \) 处最陡的正斜率。

This example illustrates the general principle: the derivative graph is a shifted and scaled version of the original only for sinusoidal functions. For most functions, there is no such simple relation.

这个例子说明了一个普遍原理：只有正弦类函数的导数图像是原函数平移和缩放后的版本。对于大多数函数，不存在这种简单关系。

10. Common Mistakes and Exam Tips | 常见错误与考试技巧

Many students confuse the sign of \( f \) with the sign of \( f’ \). Remember that \( f’ > 0 \) means \( f \) is increasing, not that \( f \) is positive. A function can be negative and still increasing, for example \( f(x) = -e^{-x} \).

许多学生混淆 \( f \) 的符号与 \( f’ \) 的符号。记住 \( f’ > 0 \) 表示 \( f \) 递增，而不是 \( f \) 为正。函数可以为负但仍递增，例如 \( f(x) = -e^{-x} \)。

Do not forget that a point where \( f'(x) = 0 \) is not always a maximum or minimum; check the sign change.
When sketching \( f’ \), pay attention to vertical asymptotes and discontinuities of \( f’ \).
Use the second derivative or the slope of \( f’ \) to verify concavity, especially at candidate inflection points.
Always label axes and key coordinates in exam sketches.

不要忘记 \( f'(x) = 0 \) 的点不一定是极大值或极小值；要检查符号变化。

绘制 \( f’ \) 时，注意 \( f’ \) 的垂直渐近线和间断点。

使用二阶导数或 \( f’ \) 的斜率来验证凹凸性，尤其是在候选拐点处。

在考试画图中始终标注坐标轴和关键坐标。

With practice, you can quickly move between the graphs of \( f \) and \( f’ \). This skill is essential for solving optimisation problems, curve sketching questions, and interpreting motion in kinematics.

通过练习，你可以快速在 \( f \) 与 \( f’ \) 的图像之间切换。这项技能对于解决优化问题、曲线绘制题以及运动学中的运动解释至关重要。

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Derivative Graphs: Sketching and Feature Analysis | 导函数图像：绘制与特征分析