Second and Higher Order Derivatives | 二阶及高阶导数求解与应用

📚 Second and Higher Order Derivatives | 二阶及高阶导数求解与应用

In calculus, the derivative of a function measures the rate of change. When we differentiate the derivative again, we obtain the second derivative, and repeating this process gives higher-order derivatives. These concepts are essential for understanding motion, curve behaviour, and approximation theory.

在微积分中，函数的导数衡量变化率。对导数再次求导得到二阶导数，重复这一过程便得到高阶导数。这些概念对于理解运动、曲线形态以及逼近理论至关重要。

1. Definitions and Notation | 定义与符号

Let y = f(x) be a differentiable function. The first derivative is denoted by f'(x) or dy/dx. The second derivative is the derivative of the first derivative:

设 y = f(x) 是可导函数。一阶导数记为 f'(x) 或 dy/dx。二阶导数是一阶导数的导数：

f”(x) = d²y/dx² = d/dx(dy/dx)

The third derivative is denoted by f”'(x) or d³y/dx³. For the nth derivative, we write f⁽ⁿ⁾(x) or dⁿy/dxⁿ.

三阶导数记为 f”'(x) 或 d³y/dx³。对于 n 阶导数，我们写为 f⁽ⁿ⁾(x) 或 dⁿy/dxⁿ。

These notations come from Leibniz, Lagrange, and Newton. In IB exams, Lagrange notation f'(x), f”(x) is most common, but Leibniz notation is essential when variables are not x and y.

这些符号分别源自莱布尼茨、拉格朗日和牛顿。在 IB 考试中，拉格朗日记号 f'(x)、f”(x) 最为常见，但当变量不是 x 和 y 时，莱布尼茨记号至关重要。

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2. Computing Second and Higher Derivatives | 二阶及高阶导数的计算

To compute a second derivative, differentiate the function once, then differentiate the result again. For example:

要计算二阶导数，先对函数求一次导，再对结果求一次导。例如：

f(x) = x⁴ + 3x² − 5x + 2, f'(x) = 4x³ + 6x − 5, f”(x) = 12x² + 6

For trigonometric functions, keep the chain rule in mind. If y = sin(2x), then y’ = 2cos(2x) and y” = −4sin(2x). Higher derivatives cycle through sine and cosine with an extra factor each time.

对于三角函数，要牢记链式法则。若 y = sin(2x)，则 y’ = 2cos(2x)，y” = −4sin(2x)。更高阶导数会在正弦与余弦之间循环，并且每次多出一个因子。

For exponential functions, y = eᵏˣ has y⁽ⁿ⁾ = kⁿeᵏˣ. For natural logarithms, y = ln x gives y’ = 1/x, y” = −1/x², and y”’ = 2/x³.

对于指数函数，y = eᵏˣ 的 n 阶导数为 y⁽ⁿ⁾ = kⁿeᵏˣ。对于自然对数，y = ln x 得到 y’ = 1/x，y” = −1/x²，y”’ = 2/x³。

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3. Leibniz Notation and Rates of Change | 莱布尼茨记号与变化率

In Leibniz notation, the second derivative is written as d²y/dx². This reads as “dee two y by dee x squared” and emphasises that differentiation is performed twice with respect to x.

在莱布尼茨记号中，二阶导数写作 d²y/dx²。这读作“d 方 y 比 d x 方”，强调对 x 进行了两次求导。

If a variable depends on time t, such as position s(t), then the first derivative ds/dt is velocity, and the second derivative d²s/dt² is acceleration. The third derivative d³s/dt³ is called jerk.

如果某个变量依赖于时间 t，例如位置 s(t)，那么一阶导数 ds/dt 是速度，二阶导数 d²s/dt² 是加速度，三阶导数 d³s/dt³ 称为加加速度（急动度）。

Leibniz notation is also useful when applying the chain rule in differential form. For example, if y = u² and u = x³ + 1, then dy/dx = (dy/du)(du/dx).

莱布尼茨记号在微分形式下应用链式法则时也非常方便。例如，若 y = u²，u = x³ + 1，则 dy/dx = (dy/du)(du/dx)。

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4. Geometric Interpretation: Concavity | 几何意义：凹凸性

The second derivative tells us how the slope itself is changing. If f”(x) > 0 on an interval, the graph is concave up (curving upward). If f”(x) < 0, the graph is concave down (curving downward).

二阶导数告诉我们斜率本身如何变化。若在某区间上 f”(x) > 0，则图像是凹向上的（向上弯曲）。若 f”(x) < 0，则图像是凹向下的（向下弯曲）。

A point where the concavity changes is called an inflection point. At such a point, f”(x) = 0 or f”(x) is undefined, and the sign of f”(x) changes across the point.

凹凸性发生改变的点称为拐点。在拐点处，f”(x) = 0 或 f”(x) 不存在，且 f”(x) 的符号在该点两侧发生改变。

For example, f(x) = x³ has f”(x) = 6x. Since f” changes from negative to positive at x = 0, the point (0,0) is an inflection point, even though f'(0) = 0.

例如，f(x) = x³ 的 f”(x) = 6x。由于 f” 在 x = 0 处由负变正，因此点 (0,0) 是拐点，尽管 f'(0) = 0。

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5. The Second Derivative Test | 二阶导数检验法

The second derivative test helps classify stationary points. Suppose f'(c) = 0. If f”(c) > 0, then f has a local minimum at c. If f”(c) < 0, then f has a local maximum at c. If f''(c) = 0, the test is inconclusive.

二阶导数检验法用于判定驻点的性质。设 f'(c) = 0。若 f”(c) > 0，则 f 在 c 处有局部极小值；若 f”(c) < 0，则 f 在 c 处有局部极大值；若 f''(c) = 0，则检验失效。

For example, consider f(x) = x³ − 3x. Then f'(x) = 3x² − 3 = 3(x−1)(x+1), so stationary points occur at x = ±1. Since f”(x) = 6x, we have f”(1) = 6 > 0 (minimum) and f”(−1) = −6 < 0 (maximum).

例如，考虑 f(x) = x³ − 3x。则 f'(x) = 3x² − 3 = 3(x−1)(x+1)，因此驻点出现在 x = ±1。由于 f”(x) = 6x，我们得到 f”(1) = 6 > 0（极小值），f”(−1) = −6 < 0（极大值）。

When f”(c) = 0, use a sign chart of the first derivative or higher-order derivatives to decide the nature of the stationary point.

当 f”(c) = 0 时，可使用一阶导数的符号表或更高阶导数来判断驻点的性质。

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6. Applications in Kinematics | 运动学中的应用

In kinematics, the position of a particle moving along a line is given by s(t). The velocity is v(t) = s'(t), and the acceleration is a(t) = s”(t).

在运动学中，沿直线运动的质点位置由 s(t) 给出。速度为 v(t) = s'(t)，加速度为 a(t) = s”(t)。

For example, if s(t) = t³ − 6t² + 9t + 2, then v(t) = 3t² − 12t + 9 and a(t) = 6t − 12. Setting a(t) = 0 gives t = 2. At this instant, the velocity may be at a maximum or minimum; indeed v'(t) = a(t).

例如，

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