A-Level Maths: How to Sketch Derivative Graphs | A-Level数学：导函数图像的绘制方法

📚 A-Level Maths: How to Sketch Derivative Graphs | A-Level数学：导函数图像的绘制方法

When studying A-Level Maths, sketching derivative graphs is an essential skill. The derivative f'(x) measures the rate of change of f(x), and its graph gives an immediate visual summary of where the original function is rising, falling, or staying flat.

在A-Level数学中，绘制导函数图像是一项必备技能。导函数 f'(x) 度量 f(x) 的变化率，其图像能让你迅速看出原函数在哪些区间上升、下降或保持水平。

1. The Relationship Between f(x) and f'(x) | f(x) 与 f'(x) 的关系

For any value of x, f'(x) is the gradient of the tangent to y = f(x). This means that the derivative graph is not a picture of the original curve; it is a picture of the slope of the original curve at every point.

对于任意 x，f'(x) 都是 y = f(x) 上切线的斜率。也就是说，导函数图像并不是原曲线的形状，而是在每个点上原曲线斜率的图像。

The table below shows the basic connection:

下表展示了最基本的关系：

2. Increasing and Decreasing Intervals | 增区间与减区间

If f is increasing on an interval, the tangent lines all slope upwards, so f'(x) > 0 everywhere in that interval. If f is decreasing, the tangent lines slope downwards, so f'(x) < 0.

如果 f 在某个区间上递增，那么所有切线都向上倾斜，因此该区间内处处有 f'(x) > 0。如果 f 递减，则切线向下倾斜，因此 f'(x) < 0。

When you sketch f'(x), translate this directly: the positive parts of the derivative graph lie above the x-axis, and the negative parts lie below it.

绘制 f'(x) 时可以直接转换：导函数图像的正值部分位于 x 轴上方，负值部分位于 x 轴下方。

3. Stationary Points and Zeros of f'(x) | 驻点与 f'(x) 的零点

A stationary point on y = f(x) occurs where f'(x) = 0. On the derivative graph, these are the x-intercepts. A local maximum or minimum is usually associated with f'(x) changing sign at that x-value.

y = f(x) 的驻点出现在 f'(x) = 0 处。在导函数图像上，这些点就是与 x 轴的交点。局部最大值或最小值通常对应 f'(x) 在该 x 值处改变正负号。

There are three common cases:

常见的情况有以下三种：

• Local maximum: f'(x) changes from positive to negative.

  局部最大值：f'(x) 由正变负。

• Local minimum: f'(x) changes from negative to positive.

  局部最小值：f'(x) 由负变正。

• Stationary inflection: f'(x) does not change sign, for example from positive to positive.

  驻点拐点：f'(x) 不改变正负号，例如由正到正。

4. Concavity and the Second Derivative | 凹凸性与二阶导数

The second derivative f”(x) describes the gradient of f'(x). Hence f”(x) > 0 means the derivative graph is increasing, while f”(x) < 0 means the derivative graph is decreasing.

二阶导数 f”(x) 描述的是 f'(x) 的斜率。因此，f”(x) > 0 说明导函数图像在上升，f”(x) < 0 说明导函数图像在下降。

In terms of the original curve:

就原曲线而言：

• f”(x) > 0: f(x) is convex, shaped like a cup.

  f”(x) > 0：f(x) 是凸函数，形状像“杯口朝上”。

• f”(x) < 0: f(x) is concave, shaped like a cap.

  f”(x) < 0：f(x) 是凹函数，形状像“帽口朝下”。

• Point of inflection: f”(x) changes sign and f'(x) has a turning point.

  拐点：f”(x) 改变正负号，f'(x) 在该处出现极值点。

5. Vertical Asymptotes and Undefined Derivatives | 垂直渐近线与导数不存在的情况

If the original function has a vertical asymptote at x = a, then the gradient often becomes infinitely large or small near a. As a result, f'(x) tends to +∞ or −∞, and the derivative graph has its own vertical asymptote.

如果原函数在 x = a 处有垂直渐近线，那么附近的斜率往往会变得非常大或非常小。因此 f'(x) 会趋向 +∞ 或 −∞，导函数图像也会有自己的垂直渐近线。

If f is not differentiable at a point, such as a corner, a cusp, or a vertical tangent, then f'(x) is undefined there. On the

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