Second Order Derivatives | 二阶导数

📚 Second Order Derivatives | 二阶导数

ary 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。

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1. Notation and Definition | 记号与定义

ve. 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。

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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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