📚 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 = dydx | 导数 = 切线斜率 = dydx
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) = x3 - 3x , we have f'(x) = 3x2 - 3 . The derivative is zero at x = ±1 .
例如：对于 f(x) = x3 - 3x ，有 f'(x) = 3x2 - 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) = x3 has f'(x) = 3x2 . 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) = x3 中 f'(x) = 3x2 。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) → 0 as x → ±∞ . 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 → ±∞ 时 f'(x) → 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) = 1x has f'(x) = -1⁄x2 , 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) = 1x 的导数为 f'(x) = -1⁄x2 ，两者均在 x=0 处有垂直渐近线。
When sketching f' , always consider the limits as x → ±∞ . This gives the overall trend and helps place the graph correctly.
绘制 f' 时，始终考虑 x → ±∞ 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) = x3 - 6x2 + 9x | 实例一：多项式 f(x) = x3 - 6x2 + 9x
Let us sketch f and its derivative f'(x) = 3x2 - 12x + 9 . First factor the derivative.
让我们绘制 f 及其导数 f'(x) = 3x2 - 12x + 9 的图像。首先对导数因式分解。
f'(x) = 3(x2 - 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 = ±π2, ±3π2, … , 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 = ±π2, ±3π2, … 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 -π2 to π2 ), 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 -π2 到 π2 ）， 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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