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。

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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。

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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）。

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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。

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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’ )。

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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。

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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。

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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。

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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。

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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{3pi}{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{3pi}{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。

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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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