Calculus: The Second Derivative – Concept, Calculation, and Significance | 微积分：二阶导数的概念、计算与意义

📚 Calculus: The Second Derivative – Concept, Calculation, and Significance | 微积分：二阶导数的概念、计算与意义

The journey of calculus begins with the first derivative, which tells us about the slope of a curve or the rate of change. But what happens when things change at an ever-changing rate? This is where the second derivative comes in.

微积分的学习始于一阶导数，它告诉我们曲线的斜率或变化率。但如果变化率本身也在不断变化，会发生什么？这时就需要引入二阶导数。

1. Defining the Second Derivative | 定义二阶导数

The second derivative is simply the derivative of the first derivative. If you differentiate a function twice, you obtain its second derivative.

二阶导数就是一阶导数的导数。如果对一个函数求两次导，就得到它的二阶导数。

Formally, for a function f(x), the second derivative, denoted as f”(x), is the derivative of f'(x).

形式上，对于函数 f(x)，记作 f”(x) 的二阶导数就是 f'(x) 的导数。

f”(x) = [f'(x)]’

The process is the same as finding the first derivative, just applied twice.

计算过程与求一阶导数相同，只是需要应用两次。

2. Notation for the Second Derivative | 二阶导数的常见记号

There are several standard notations for the second derivative. Understanding them is crucial since different textbooks and exam boards use different forms.

二阶导数有几种标准记号。理解它们至关重要，因为不同的教材和考试局会使用不同的形式。

Lagrange notation (拉格朗日记号): f”(x) or f”
Leibniz notation (莱布尼茨记号): d²y/dx²
Newton notation (牛顿记号): ÿ (mainly used in physics for time derivatives)

The Leibniz notation reads as “d squared y by d x squared” and emphasizes the operation of differentiation applied twice.

莱布尼茨记号读作”d²y 比 dx²”，强调进行两次微分运算。

3. Computing Basic Second Derivatives | 基础二阶导数的计算

Let’s calculate the second derivative of a simple power function to see the process in action.

让我们计算一个简单幂函数的二阶导数，实地看一下这个过程。

Consider f(x) = x³. The first derivative is f'(x) = 3x². Differentiating again yields f”(x) = 6x.

考虑 f(x) = x³。一阶导数为 f'(x) = 3x²。再次求导得到 f”(x) = 6x。

The key is applying the power rule sequentially: bring the power down, multiply, and reduce the exponent by one.

关键在于连续应用幂法则：将指数下移相乘，然后将指数减一。

4. Second Derivatives of Polynomials | 多项式函数的二阶导数

Let’s explore a more complex polynomial. If f(x) = 4x⁴ – 3x² + 2x – 7, then f'(x) = 16x³ – 6x + 2. Applying the rule again, f”(x) = 48x² – 6.

让我们探索一个更复杂的多项式。如果 f(x) = 4x⁴ – 3x² + 2x – 7，那么 f'(x) = 16x³ – 6x + 2。再次应用法则，f”(x) = 48x² – 6。

Notice how the constant term -7 vanished after the first derivative, and the linear term 2x became a constant before vanishing in the second derivative.

请注意，常数项 -7 在一阶导数后消失，线性项 2x 在二阶导数中变成常数后也消失了。

5. Second Derivatives of Trigonometric Functions | 三角函数的二阶导数

Trigonometric functions exhibit cyclical behaviour in their derivatives. For example, if f(x) = sin(x), then f'(x) = cos(x), and f”(x) = -sin(x).

三角函数的导数具有周期性。例如，如果 f(x) = sin(x)，那么 f'(x) = cos(x)，f”(x) = -sin(x)。

Differentiating f(x) = cos(x) gives f'(x) = -sin(x), so f”(x) = -cos(x).

对 f(x) = cos(x) 求导得到 f'(x) = -sin(x)，因此 f”(x) = -cos(x)。

The negative sign is essential! It indicates that the original function is being scaled by -1 after two differentiations.

负号至关重要！它表明原函数在两次求导后乘以了 -1。

6. Second Derivatives of Exponential and Logarithmic Functions | 指数与对数函数的二阶导数

The exponential function is unique in that its derivatives are itself. If f(x) = eˣ, then f'(x) = eˣ and f”(x) = eˣ.

指数函数很独特，它的导数就是它本身。如果 f(x) = eˣ，那么 f'(x) = eˣ，f”(x) = eˣ。

For natural logarithms, if f(x) = ln(x), then f'(x) = 1/x = x⁻¹. Using the power rule, f”(x) = -x⁻² = -1/x².

对于自然对数，如果 f(x) = ln(x)，那么 f'(x) = 1/x = x⁻¹。使用幂法则，f”(x) = -x⁻² = -1/x²。

Be careful with the domain: the second derivative of ln(x) is negative for all x in its domain (x > 0).

注意定义域：在 ln(x) 的定义域内（x > 0），其二阶导数始终为负。

7. Implicit Differentiation for Second Derivatives | 隐函数求二阶导

Sometimes a function is not given explicitly, but implicitly, like x² + y² = 25. Differentiating implicitly with respect to x gives 2x + 2y(dy/dx) = 0.

有时函数不是显式给出的，而是隐式给出的，例如 x² + y² = 25。对 x 隐式求导得到 2x + 2y(dy/dx) = 0。

Solving for dy/dx gives -x/y. To find d²y/dx², we differentiate -x/y with respect to x using the quotient rule.

解出 dy/dx 得 -x/y。为求 d²y/dx²，我们使用商法则对 -x/y 关于 x 求导。

d²y/dx² = -(y – x(dy/dx)) / y²

This yields -(y – x(dy/dx)) / y². Substituting dy/dx = -x/y gives -(y + x²/y) / y² = -(y² + x²) / y³.

这得到 -(y – x(dy/dx)) / y²。代入 dy/dx = -x/y 得 -(y + x²/y) / y² = -(y² + x²) / y³。

Since x² + y² = 25, the second derivative simplifies to -25 / y³.

由于 x² + y² = 25，二阶导数简化为 -25 / y³。

8. Second Derivatives of Parametric Equations | 参数方程的二阶导数

Parametric equations define x and y separately in terms of a parameter, often t. For example, x = t² and y = t³. We know dy/dx = (dy/dt) / (dx/dt) = 3t² / 2t = (3/2)t.

参数方程用参数（通常是 t）分别定义 x 和 y。例如，x = t²，y = t³。我们知道 dy/dx = (dy/dt) / (dx/dt) = 3t² / 2t = (3/2)t。

To find the second derivative, we cannot simply differentiate dy/dx with respect to t. We must differentiate dy/dx with respect to x.

为了求二阶导数，我们不能简单地关于 t 对 dy/dx 求导。必须关于 x 对 dy/dx 求导。

d²y/dx² = [d/dt(dy/dx)] / (dx/dt)

The formula is d²y/dx² = (d/dt(dy/dx)) / (dx/dt). Here, d/dt((3/2)t) = 3/2. Dividing by dx/dt = 2t gives 3 / (4t).

公式是 d²y/dx² = (d/dt(dy/dx)) / (dx/dt)。这里，d/dt((3/2)t) = 3/2。除以 dx/dt = 2t 得到 3 / (4t)。

9. Physical Meaning: Acceleration | 物理意义：加速度

In physics, the second derivative has a profound interpretation. If s(t) represents the position of an object moving in a straight line at time t, then the first derivative s'(t) is the velocity, and the second derivative s”(t) is the acceleration.

在物理学中，二阶导数有着深刻的含义。如果 s(t) 表示物体在时间 t 沿直线运动的位置，那么一阶导数 s'(t) 是速度，二阶导数 s”(t) 是加速度。

Acceleration is the rate at which velocity changes. It is the “slope” of the velocity-time graph.

加速度是速度变化的速率。它是速度-时间图像上的”斜率”。

This concept extends to many other fields. In economics, the second derivative can measure the rate of change of marginal cost.

这个概念可以扩展到许多其他领域。在经济学中，二阶导数可以衡量边际成本的变化率。

10. Geometric Meaning: Concavity | 几何意义：凹凸性

The second derivative tells us about the concavity of a curve. If f”(x) > 0 on an interval, the curve is concave up (shaped like a cup, ∪).

二阶导数告诉我们曲线的凹凸性。如果在某区间上 f”(x) > 0，曲线是凹向上的（形如杯子，∪）。

If f”(x) < 0, the curve is concave down (shaped like a cap, ∩). The curve lies below its tangents when concave down, and above them when concave up.

如果 f”(x) < 0，曲线是凸向上的（形如帽子，∩）。当凸向上时，曲线位于其切线之下；当凹向上时，位于切线之上。

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

凹凸性发生改变的点称为拐点。在这些点上，f”(x) = 0 或不存在。

11. The Second Derivative Test | 二阶导数判别法

The second derivative provides a powerful technique to identify local maxima and minima, known as the Second Derivative Test.

二阶导数提供了一种判定局部极大值和极小值的有效方法，称为二阶导数判别法。

Suppose x = c is a stationary point, meaning f'(c) = 0. If f”(c) > 0, the function has a local minimum at c. If f”(c) < 0, it has a local maximum. If f''(c) = 0, the test is inconclusive.

设 x = c 是驻点，即 f'(c) = 0。如果 f”(c) > 0，函数在 c 处有局部极小值。如果 f”(c) < 0，它有局部极大值。如果 f''(c) = 0，则判别法失效。

This test is often quicker than constructing a sign table for the first derivative, making it invaluable in exam settings.

这个判别法通常比为一阶导数构建符号表更快，在考试环境中非常有价值。

12. Summary | 总结

The second derivative is a fundamentally important concept in calculus. It measures the rate of change of the first derivative, providing insights into a function’s shape, its concavity, and the nature of its stationary points.

二阶导数是微积分中一个极其重要的概念。它衡量一阶导数的变化率，深入揭示了函数的形状、凹凸性及其驻点的性质。

Whether used to calculate acceleration in physics or to locate maxima in economics, mastering the computation and interpretation of the second derivative is essential for advanced mathematics.

无论是用于计算物理学中的加速度，还是用于经济学中寻找最大值，掌握二阶导数的计算与解释对于学习高等数学都至关重要。

Practice differentiating a variety of functions, from simple powers to implicit and parametric forms, to build a solid foundation.

练习对各种函数求导，从简单的幂函数到隐函数和参数方程，以打下坚实的基础。

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

Calculus: The Second Derivative – Concept, Calculation, and Significance | 微积分：二阶导数的概念、计算与意义