Second Order Derivatives | 二阶导数

📚 Second Order Derivatives | 二阶导数

Second order derivatives appear throughout A-level Mathematics, especially in curve sketching, optimisation and kinematics. They tell you not just how a function is changing, but how the gradient itself is changing. This article covers notation, calculation, classification of stationary points, points of inflection, parametric and implicit second derivatives, and common exam pitfalls.

二阶导数在 A-level 数学中随处可见，尤其是在函数图像、优化问题和运动学中。它不仅告诉你函数如何变化，还告诉你斜率本身如何变化。本文涵盖二阶导数的记号、计算、驻点分类、拐点、参数方程和隐函数的二阶导数，以及考试中常见的陷阱。

1. What is a second order derivative? | 什么是二阶导数

When you differentiate y = f(x) once, you obtain the first derivative f'(x) = dy/dx, which measures the rate of change of y with respect to x. If you differentiate the first derivative again, you get the second order derivative, written f”(x) or d²y/dx². It measures the rate at which the gradient itself is changing. In other words, the second derivative is the derivative of the derivative.

对 y = f(x) 求一次导，得到一阶导数 f'(x) = dy/dx，它衡量 y 关于 x 的变化率。再对一阶导数求导，就得到二阶导数，记作 f”(x) 或 d²y/dx²。它衡量斜率本身的变化快慢。换句话说，二阶导数就是导数的导数。

Higher order derivatives follow the same idea: the third derivative f”'(x) is the derivative of f”(x), and so on. At A-level, the second derivative is the most commonly used higher derivative, especially for identifying maxima, minima and acceleration.

更高阶的导数遵循相同的思路：三阶导数 f”'(x) 是 f”(x) 的导数，以此类推。在 A-level 中，二阶导数是最常用的高阶导数，尤其是在判断极大值、极小值和加速度时。

2. Notation and terminology | 记号与术语

The second derivative can be written in several equivalent ways. If y = f(x), then f”(x), y”, d²y/dx² and d/dx(dy/dx) all mean the same thing. The notation d²y/dx² comes from applying the operator d/dx twice. It is read as “d two y by d x squared” or “the second derivative of y with respect to x”.

二阶导数有几种等价写法。若 y = f(x)，则 f”(x)、y”、d²y/dx² 和 d/dx(dy/dx) 含义完全相同。记号 d²y/dx² 源于两次使用微分算子 d/dx。它读作 “d two y by d x squared” 或 “y 对 x 的二阶导数”。

Be careful: d²y/dx² is not the same as (dy/dx)². The first is the second derivative, while the second is the square of the first derivative. Confusing these two is a very common exam error.

注意：d²y/dx² 与 (dy/dx)² 不同。前者是二阶导数，后者是一阶导数的平方。混淆这两个记号是考试中非常常见的错误。

3. Calculating second derivatives | 计算二阶导数

To find f”(x), first find f'(x), then differentiate again. For example, if y = 3x⁴ − 5x² + 2x − 7, then dy/dx = 12x³ − 10x + 2 and d²y/dx² = 36x² − 10. The same differentiation rules apply at each step: the power rule, product rule, quotient rule and chain rule.

求 f”(x) 要先求 f'(x)，然后再求一次导。例如，若 y = 3x⁴ − 5x² + 2x − 7，则 dy/dx = 12x³ − 10x + 2，d²y/dx² = 36x² − 10。每一步都使用相同的求导法则：幂法则、乘法法则、除法法则和链式法则。

For standard functions, you can remember the common second derivatives. If y = xⁿ, then f”(x) = n(n−1)xⁿ⁻². If y = sin x, then f”(x) = −sin x. If y = cos x, then f”(x) = −cos x. If y = eˣ, then f”(x) = eˣ. If y = ln x, then f”(x) = −1/x². These results come directly from differentiating twice.

对于基本初等函数，可以记住常见的二阶导数。若 y = xⁿ，则 f”(x) = n(n−1)xⁿ⁻²。若 y = sin x，则 f”(x) = −sin x。若 y = cos x，则 f”(x) = −cos x。若 y = eˣ，则 f”(x) = eˣ。若 y = ln x，则 f”(x) = −1/x²。这些结果都是直接求两次导得到的。

4. Second derivative and concavity | 二阶导数与凹凸性

The sign of f”(x) tells you how the curve bends. If f”(x) > 0 on an interval, the gradient is increasing, and the curve is convex, also called concave up. It looks like a cup ∪ and lies above its tangents. If f”(x) < 0 on an interval, the gradient is decreasing, and the curve is concave, also called concave down. It looks like a cap ∩ and lies below its tangents.

f”(x) 的符号说明曲线的弯曲方式。若在某个区间内 f”(x) > 0，则斜率递增，曲线是凸的，也叫凹向上，形状像杯子 ∪，位于切线上方。若在某个区间内 f”(x) < 0，则斜率递减，曲线是凹的，也叫凹向下，形状像帽子 ∩，位于切线下方。

This is particularly useful when sketching graphs. A convex section has an increasing slope, while a concave section has a decreasing slope. At a point where concavity changes, you may have a point of inflection, which will be discussed later.

这在绘制函数图像时特别有用。凸区间的斜率递增，凹区间的斜率递减。在凹凸性发生改变的点，可能出现拐点，后文会详细讨论。

5. Stationary points and the second derivative test | 驻点与二阶导数检验

A stationary point occurs where f'(x) = 0. The second derivative test uses the sign of f”(x) at that point to classify the stationary point. If f'(a) = 0 and f”(a) > 0, the point is a local minimum because the gradient changes from negative to positive. If f'(a) = 0 and f”(a) < 0, the point is a local maximum because the gradient changes from positive to negative.

驻点出现在 f'(x) = 0 处。二阶导数检验利用该点处 f”(x) 的符号来判断驻点类型。若 f'(a) = 0 且 f”(a) > 0，该点是局部极小值，因为斜率由负变正。若 f'(a) = 0 且 f”(a) < 0，该点是局部极大值，因为斜率由正变负。

If f”(a) = 0, the test is inconclusive. This does not automatically mean there is a point of inflection. You should use the first derivative test by examining the sign of f'(x) on either side of a. A sign table is often the safest method in this situation.

如果 f”(a) = 0，二阶导数检验无法判断。这并不自动意味着存在拐点。此时应使用一阶导数检验，检查 a 两侧 f'(x) 的符号。在这种情况下，符号表通常是最稳妥的方法。

6. Worked example: classifying stationary points | 例题：驻点分类

Find and classify the stationary points of y = x³ − 3x + 2. First, dy/dx = 3x² − 3. Set dy/dx = 0: 3x² − 3 = 0, so x² = 1, giving x = −1 or x = 1. The second derivative is d²y/dx² = 6x.

求 y = x³ − 3x + 2 的驻点并分类。首先，dy/dx = 3x² − 3。令 dy/dx = 0：3x² − 3 = 0，所以 x² = 1，得到 x = −1 或 x = 1。二阶导数为 d²y/dx² = 6x。

At x = 1, f”(1) = 6 > 0, so there is a local minimum at (1,

Published by TutorHao | A-Level Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）