Differentiation from First Principles | 从第一原理求导

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

In Edexcel A Level Mathematics, differentiation from first principles is the foundational method that explains why the derivative rules work. It builds the derivative from the limit of a chord’s gradient, rather than simply presenting the power rule as a given. Mastering this technique is essential for high marks on the Pure Mathematics papers, especially in questions that explicitly ask for first-principles proof.

在爱德思 A-Level 数学中，从第一原理求导是解释为什么导数法则成立的基础方法。它通过弦的斜率的极限来构造导数，而不是直接把幂法则当作既定结论。掌握这一技巧对于在纯数试卷中取得高分至关重要，尤其是在明确要求用第一原理证明的问题中。

1. What Is Differentiation from First Principles? | 什么是第一原理求导？

Differentiation from first principles calculates the derivative of a function at a point by taking the limit of the gradient of a chord joining two nearby points on the curve. The derivative f'(x) represents the instantaneous rate of change or the gradient of the tangent to the curve y=f(x) at a given point. Edexcel often tests this in the context of quadratic and cubic functions.

从第一原理求导通过取曲线上两个相邻点之间的弦的斜率的极限，来计算函数在某一点的导数。导数 f'(x) 表示瞬时变化率，即曲线 y=f(x) 在给定点处切线的斜率。爱德思经常在二次函数和三次函数的背景下考查这一内容。

2. The Limit Definition of the Derivative | 导数的极限定义

The formal definition of the derivative is written as f'(x) = lim(h→0) [f(x+h) – f(x)] / h, where h is a small change in x. This expression measures the average rate of change over the interval from x to x+h, and then lets h approach zero to find the rate of change at a single point. In Edexcel exams, you must be able to write this definition clearly and use it algebraically.

导数的正式定义写作 f'(x) = lim(h→0) [f(x+h) – f(x)] / h，其中 h 是 x 的一个微小变化。该表达式测量从 x 到 x+h 区间上的平均变化率，然后让 h 趋近于零，以求出单点处的变化率。在爱德思考试中，你必须能清晰地写出这一定义，并用代数方法使用它。

f'(x) = lim(h→0) [f(x+h) – f(x)] / h

3. The Gradient of a Chord | 弦的斜率

For a small change h, the two points on the curve are (x, f(x)) and (x+h, f(x+h)). The gradient of the chord joining these points is the quotient [f(x+h) – f(x)] / h. This gives the average slope between the two points. As h becomes very small, the chord becomes a better and better approximation to the tangent at the point.

对于微小的变化 h，曲线上的两个点是 (x, f(x)) 和 (x+h, f(x+h))。连接这两点的弦的斜率就是商 [f(x+h) – f(x)] / h。这给出了两点之间斜率的平均值。随着 h 变得非常小，弦会越来越接近该点处的切线。

4. The Tangent as a Limit | 极限下的切线

The tangent to the curve at x is the limiting position of the chord as h tends to zero. Therefore, the derivative f'(x) is the limit of the chord gradient when h→0. This limiting process is the core idea of differentiation from first principles. If the limit exists, the function is said to be differentiable at that point.

曲线在 x 处的切线是弦在 h 趋近于零时的极限位置。因此，导数 f'(x) 就是当 h→0 时弦斜率的极限。这一极限过程是从第一原理求导的核心思想。如果极限存在，则函数在该点处可导。

5. Worked Example: f(x)=x² | 示例：f(x)=x²

Let f(x)=x². Then f(x+h)=(x+h)² = x² + 2xh + h², so the difference is f(x+h) – f(x) = 2xh + h². Dividing by h gives [2xh + h²] / h = 2x + h. Taking the limit as h→0 leaves 2x. This proves that the derivative of x² is 2x.

设 f(x)=x²。那么 f(x+h)=(x+h)² = x² + 2xh + h²，因此差为 f(x+h) – f(x) = 2xh + h²。除以 h 得到 [2xh + h²] / h = 2x + h。当 h→0 取极限，得到 2x。这证明了 x² 的导数是 2x。

f'(x) = lim(h→0) (2x + h) = 2x

6. Worked Example: f(x)=x³ | 示例：f(x)=x³

For f(x)=x³, expand (x+h)³ = x³ + 3x²h + 3xh² + h³. Subtracting f(x) gives 3x²h + 3xh² + h³. Dividing by h yields 3x² + 3xh + h². As h→0, both 3xh and h² tend to zero, so the limit is 3x². Thus the derivative of x³ is 3x².

对于 f(x)=x³，展开 (x+h)³ = x³ + 3x²h + 3xh² + h³。减去 f(x) 得到 3x²h + 3xh² + h³。除以 h 得到 3x² + 3xh + h²。当 h→0 时，3xh 和 h² 都趋于零，所以极限是 3x²。因此 x³ 的导数是 3x²。

f'(x) = lim(h→0) (3x² + 3xh + h²) = 3x²

7. The General Power Rule Connection | 与幂法则的联系

From these examples, a pattern emerges: for f(x)=xⁿ, the first-principles expansion of (x+h)ⁿ produces a leading term n xⁿ⁻¹h after subtraction, followed by terms containing higher powers of h. After dividing by h and letting h→0, only n xⁿ⁻¹ remains. This explains why the power rule f'(x)=n xⁿ⁻¹ is valid.

从这些例子中可以看出一个模式：对于 f(x)=xⁿ，第一原理展开 (x+h)ⁿ 后在减去 f(x) 时会产生一个首项 n xⁿ⁻¹h，其余项都含有 h 的更高次幂。除以 h 并让 h→0 后，只剩下 n xⁿ⁻¹。这就解释了为什么幂法则 f'(x)=n xⁿ⁻¹ 成立。

8. Common Algebraic Pitfalls | 常见代数陷阱

A common mistake is to write (x+h)² as x² + h², forgetting the cross term 2xh. Another error is to cancel h incorrectly before expanding the numerator. In the limit step, students sometimes substitute h=0 too early, which gives a 0/0 form. Always simplify the quotient fully before taking the limit.

一个常见错误是把 (x+h)² 写成 x² + h²，遗漏了交叉项 2xh。另一个错误是在展开分子之前错误地约去 h。在极限步骤中，学生有时过早代入 h=0，导致 0/0 的形式。务必在取极限之前先完全化简该商。

9. Applying the Definition to Other Functions | 应用到其他函数

The same definition can be applied to functions such as f(x)=1/x or f(x)=√x, but the algebra is more demanding. For rational functions, combine fractions over a common denominator. For square roots, multiply by the conjugate. Edexcel A-Level questions usually focus on polynomial examples, but you should understand how the limit process generalises.

同样的定义也适用于其他函数，例如 f(x)=1/x 或 f(x)=√x，但代数运算要求更高。对于有理函数，要通分合并分数。对于平方根，应乘以共轭式。爱德思 A-Level 考题通常集中在多项式示例上，但你应该理解极限过程如何推广。

10. Exam Technique and Marking Points | 考试技巧与得分点

In an exam, show all four steps clearly: write the definition, substitute f(x+h) and f(x), simplify the difference quotient, and state the limit as h→0. Even if the final answer is obvious, you must display the full limit process to gain method marks. Use correct notation such as f'(x) and lim(h→0) throughout.

在考试中，要清楚地写出四个步骤：写出定义，代入 f(x+h) 和 f(x)，化简差商，并写出 h→0 时的极限。即使最终答案很明显，也必须展示完整的极限过程才能获得方法分。始终使用正确的符号，如 f'(x) 和 lim(h→0)。

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Differentiation from First Principles | 从第一原理求导