Differentiation from First Principles | 从第一原理求导

📚 Differentiation from First Principles | 从第一原理求导

Differentiation from first principles is the foundation of A-Level calculus. It defines the derivative as the limit of an average rate of change over a vanishingly small interval. On Edexcel papers, you may be asked to prove a derivative from first principles or to interpret what the derivative means.

从第一原理求导是 A-Level 微积分的基础。它把导数定义为在趋近于零的区间上平均变化率的极限。在 Edexcel 考试中，你可能会被要求用第一原理证明某个导数，或解释导数的含义。

1. The Meaning of a Derivative | 导数的含义

A derivative measures how quickly a function changes as its input changes. At a point on a curve, the derivative gives the gradient of the tangent to the curve. This is also called the instantaneous rate of change.

导数衡量函数值随自变量变化而变化的快慢。在曲线上某一点，导数给出该点切线的斜率。这也称为瞬时变化率。

derivative = gradient of the tangent = instantaneous rate of change

导数 = 切线斜率 = 瞬时变化率

2. Secants and Tangents | 割线与切线

To find the gradient of a curve at a point, we start with a secant line joining two nearby points. If the two points have x-coordinates x and x + h, the secant gradient is [f(x+h) − f(x)] / h. As h approaches zero, the secant line becomes the tangent line at x.

为了求曲线上某点的斜率，我们先取连接两个邻近点的割线。如果两个点的横坐标为 x 和 x + h，则割线斜率为 [f(x+h) − f(x)] / h。当 h 趋于零时，割线变成 x 处的切线。

3. The First Principles Formula | 第一原理公式

The derivative of f(x) is written as f'(x). The formal definition is given by the limit of the difference quotient:

函数 f(x) 的导数记作 f'(x)。正式定义由差商的极限给出：

f'(x) = lim (h → 0) [f(x+h) − f(x)] / h

This limit expresses the idea that we let the change h become vanishingly small. The expression [f(x+h) − f(x)] / h is the average rate of change over the interval from x to x + h.

这个极限表示让变化量 h 趋近于零的思想。表达式 [f(x+h) − f(x)] / h 是从 x 到 x + h 区间上的平均变化率。

4. Worked Example: f(x) = x² | 例题：f(x) = x²

Let f(x) = x². Then f(x+h) = (x+h)² = x² + 2xh + h². Subtract f(x) to get 2xh + h², and divide by h:

设 f(x) = x²，则 f(x+h) = (x+h)² = x² + 2xh + h²。减去 f(x) 得到 2xh + h²，再除以 h：

[f(x+h) − f(x)] / h = (2xh + h²) / h = 2x + h

Taking the limit as h → 0 gives f'(x) = 2x. This confirms that the gradient of y = x² at any point is 2x.

令 h → 0 取极限，得到 f'(x) = 2x。这证实了 y = x² 在任意点的斜率为 2x。

5. Worked Example: f(x) = x³ | 例题：f(x) = x³

Let f(x) = x³. Expanding (x+h)³ gives x³ + 3x²h + 3xh² + h³. Subtract x³ and divide by h:

设 f(x) = x³，展开 (x+h)³ 得 x³ + 3x²h + 3xh² + h³。减去 x³ 再除以 h：

[f(x+h) − f(x)] / h = (3x²h + 3xh² + h³) / h = 3x² + 3xh + h²

As h tends to zero, the terms 3xh and h² vanish, so f'(x) = 3x². The same method can be used for higher powers of x.

当 h 趋于零时，3xh 和 h² 项消失，因此 f'(x) = 3x²。同样的方法可用于更高次幂的 x。

6. The Power Rule | 幂函数求导法则

From these results a pattern emerges: if y = xⁿ, then dy/dx = nxⁿ⁻¹. This is the power rule, and it works for any real constant n in the Edexcel A-Level syllabus.

从以上结果可以归纳出规律：如果 y = xⁿ，则 dy/dx = nxⁿ⁻¹。这就是幂函数求导法则，在 Edexcel A-Level 大纲中适用于任意实常数 n。

d/dx (xⁿ) = n xⁿ⁻¹

For example, if y = x⁵, then dy/dx = 5x⁴. If y = x⁰.⁵, then dy/dx = 0.5x⁻⁰.⁵. This rule saves time compared with using the limit definition every time.

例如，如果 y = x⁵，则 dy/dx = 5x⁴。如果 y = x⁰.⁵，则 dy/dx = 0.5x⁻⁰.⁵。与每次使用极限定义相比，该法则节省了时间。

7. Constant and Multiple Rules | 常数与倍数法则

The derivative of a constant is zero: d/dx(c) = 0. If a function is multiplied by a constant, the constant can be taken outside the derivative: d/dx[k f(x)] = k f'(x).

常数的导数为零：d/dx(c) = 0。如果一个函数乘以常数，常数可以提到导数符号外：d/dx[k f(x)] = k f'(x)。

d/dx (5x⁴) = 5 × 4x³ = 20x³

These rules are assumed in later differentiation topics, including stationary points, tangents and normals, and optimisation.

这些法则在后续的微分主题中都会用到，包括驻点、切线与法线以及优化问题。

8. Sum and Difference Rule | 和差法则

You can differentiate term by term. For u = f(x) and v = g(x), d/dx(u + v) = du/dx + dv/dx, and d/dx(u − v) = du/dx − dv/dx.

可以逐项求导。设 u = f(x)、v = g(x)，则 d/dx(u + v) = du/dx + dv/dx，d/dx(u − v) = du/dx − dv/dx。

d/dx (x³ + 4x² − 2x + 7) = 3x² + 8x − 2

The constant term 7 differentiates to zero. This rule is essential for handling polynomial functions and many functions seen in the A-Level Pure Mathematics papers.

常数项 7 求导为零。这个法则对于处理多项式函数以及 A-Level 纯数试卷中出现的许多函数都至关重要。

9. Tangents and Normals | 切线与法线

Once dy/dx is known, the gradient of the tangent at x = a is m = f'(a). The normal is perpendicular to the tangent, so its gradient is −1/m. Use y − y₁ = m(x − x₁) to write the line equation.

求出 dy/dx 后，x = a 处切线的斜率为 m = f'(a)。法线垂直于切线，因此其斜率为 −1/m。用 y − y₁ = m(x − x₁) 写出直线方程。

gradient of normal = −1 / gradient of tangent

法线斜率 = −1 / 切线斜率

Many Edexcel questions ask for the equation of a tangent or normal at a given point. Always work out the y-coordinate first if it is not given.

许多 Edexcel 题目要求求给定点处的切线或法线方程。如果未给出 y 坐标，一定要先求出它。

10. The Second Derivative | 二阶导数

Differentiating twice gives the second derivative, written as f”(x) or d²y/dx². It describes the rate of change of the gradient and helps determine whether a function is concave up or concave down.

对导数再求导得到二阶导数，记作 f”(x) 或 d²y/dx²。它描述斜率的变化率，有助于判断函数是上凹还是下凹。

y = x⁴ → dy/dx = 4x³ → d²y/dx² = 12x²

In mechanics, the second derivative of displacement with respect to time is acceleration. In pure mathematics, the sign of f”(x) can classify stationary points.

在力学中，位移对时间的二阶导数是加速度。在纯数学中，f”(x) 的符号可用于对驻点进行分类。

11. Common Mistakes | 常见错误

Students often forget to write the limit step in a first principles proof. Another mistake is expanding brackets incorrectly when finding f(x+h). Also, when finding a normal, many candidates use gradient m instead of −1/m.

学生经常忘记在第一原理证明中写出极限步骤。另一个错误是求 f(x+h) 时展开括号出错。此外，求法线时许多考生错用了斜率 m 而不是 −1/m。

Always write lim (h → 0) before taking the limit.

在取极限之前一定要写出 lim (h → 0)。
Simplify [f(x+h) − f(x)] / h fully before letting h tend to 0.

在让 h 趋于 0 之前，先完全化简 [f(x+h) − f(x)] / h。
For a normal, check that the product of the two gradients is −1.

对于法线，检查两个斜率的乘积是否为 −1。

12. Exam Tips | 考试技巧

In a ‘show from first principles’ question, write the definition, expand f(x+h), simplify the quotient, and then state that h → 0. Always simplify fully before taking the limit. For tangent or normal questions, give exact answers unless the question states otherwise.

在 ‘用第一原理证明’ 的题目中，写出定义、展开 f(x+h)、化简差商，然后说明 h → 0。在取极限前一定要完全化简。对于切线或法线问题，除非题目另有说明，答案保留精确值。

If a question says ‘you may assume the derivative of xⁿ is nxⁿ⁻¹’, do not start from first principles unless asked. Show enough working to make your reasoning clear, especially when a substitution or coordinate calculation is involved.

如果题目说 ‘你可以假设 xⁿ 的导数是 nxⁿ⁻¹’，除非要求，否则不要从第一原理开始。展示足够的过程使推理清晰，尤其是在涉及代入或坐标计算时。

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Differentiation from First Principles | 从第一原理求导