Second Order Derivatives | 二阶导数

📚 Second Order Derivatives | 二阶导数

In differential calculus, the second order derivative measures the rate of change of the first derivative. It reveals how the gradient itself is changing, giving insight into curvature, acceleration, and the nature of stationary points. For a function y = f(x), the second derivative is written as f″(x) or d²y/dx².

在微分学中，二阶导数度量的是导数的变化率。它揭示了梯度本身如何变化，为我们提供了关于曲线弯曲程度、加速度以及驻点性质的深刻洞察。对于函数 y = f(x)，二阶导数记为 f″(x) 或 d²y/dx²。

1. Notation and Definition | 记号与定义

The second derivative is obtained by differentiating the first derivative. If y = f(x), then the first derivative is dy/dx = f′(x), and the second derivative is:

二阶导数是对一阶导数再次求导的结果。若 y = f(x)，则一阶导数为 dy/dx = f′(x)，二阶导数为：

d²y/dx² = f″(x) = d/dx (dy/dx)

Several notations are in common use:

常用的记号有：

• Leibniz notation: d²y/dx²
• 莱布尼茨记号：d²y/dx²
• Prime notation: f″(x) or y″
• 撇号记号：f″(x) 或 y″
• Operator notation: D²f(x)
• 算子记号：D²f(x)

It is essential to recognise that d²y/dx² means d(dy/dx)/dx, not (dy/dx)². The superscript 2 in the denominator indicates differentiation performed twice with respect to x.

必须明确的是，d²y/dx² 表示 d(dy/dx)/dx，而不是 (dy/dx)²。分母中的上标 2 表示对 x 进行了两次求导。

2. Finding Second Derivatives | 求二阶导数

To compute the second derivative, differentiate the first derivative using the standard rules. Consider the following examples:

求二阶导数时，只需运用标准求导法则对一阶导数再求导。请看以下例子：

Example 1. Let f(x) = x⁵ − 3x² + 7x − 2.

例 1。设 f(x) = x⁵ − 3x² + 7x − 2。

First derivative: f′(x) = 5x⁴ − 6x + 7.

一阶导数：f′(x) = 5x⁴ − 6x + 7。

Second derivative: f″(x) = 20x³ − 6.

二阶导数：f″(x) = 20x³ − 6。

Example 2. Let y = sin(2x). Then dy/dx = 2 cos(2x), and d²y/dx² = −4 sin(2x).

例 2。设 y = sin(2x)，则 dy/dx = 2 cos(2x)，且 d²y/dx² = −4 sin(2x)。

Example 3. Let y = e³ˣ. Then dy/dx = 3e³ˣ and d²y/dx² = 9e³ˣ, following the pattern d²(eᵏˣ)/dx² = k²eᵏˣ.

例 3。设 y = e³ˣ，则 dy/dx = 3e³ˣ，二阶导数为 d²y/dx² = 9e³ˣ，符合规律 d²(eᵏˣ)/dx² = k²eᵏˣ。

For products and quotients, use the product or quotient rule first, then differentiate the resulting expression.

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