Differentiation from First Principles | 第一原理求导

📚 Differentiation from First Principles | 第一原理求导

Differentiation from first principles is one of the most important foundational skills in Edexcel A-Level Pure Mathematics. It asks you to find the gradient of a curve directly from the limit definition of the derivative, rather than simply applying differentiation rules. This topic appears regularly in Paper 1 and requires careful algebraic manipulation, clear use of limit notation, and a solid understanding of what a derivative actually measures.

第一原理求导是爱德思 A-Level 纯数中最重要的基础技能之一。它要求你直接从导数的极限定义出发求曲线斜率，而不是直接套用求导法则。这个主题在 Paper 1 中经常出现，需要细致的代数处理、清晰的极限符号书写，以及对导数本质含义的扎实理解。

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

For a function f(x), the derivative f ‘(x) is defined as the limit of the average rate of change of f over an interval of length h, as h tends to zero. This limit gives the instantaneous rate of change of f with respect to x.

对于函数 f(x)，导数 f ‘(x) 定义为当区间长度 h 趋于零时，f 在该区间上的平均变化率的极限。这个极限给出了 f 关于 x 的瞬时变化率。

f ‘(x) = limₕ→₀ [f(x + h) − f(x)] / h

The expression inside the limit is the gradient of the secant line joining the points (x, f(x)) and (x + h, f(x + h)). As h approaches zero, the secant line becomes the tangent line at x, and its gradient becomes f ‘(x).

极限号内的表达式是连接点 (x, f(x)) 和点 (x + h, f(x + h)) 的割线斜率。当 h 趋近于零时，割线逐渐变成点 x 处的切线，其斜率就是 f ‘(x)。

2. Geometrical Interpretation | 几何意义

Geometrically, differentiation from first principles is a limiting process. Start with a chord between two nearby points on the curve y = f(x). The chord has gradient [f(x + h) − f(x)] / h. As h becomes smaller, the chord shortens and rotates until it becomes the tangent at the single point x.

从几何上看，第一原理求导是一个极限过程。先在曲线 y = f(x) 上取两个邻近点，连接成一条弦。这条弦的斜率为 [f(x + h) − f(x)] / h。随着 h 变小，弦不断缩短并旋转，最终变成点 x 处的切线。

This is why the derivative is often called the gradient function. It gives the slope of the tangent for every value of x, provided the limit exists.

这就是为什么导数通常被称为梯度函数。只要极限存在，它就能给出每一个 x 值对应的切线斜率。

3. Differentiating f(x) = x² from First Principles | 从第一原理求 f(x) = x² 的导数

To differentiate f(x) = x² from first principles, first write f(x + h) = (x + h)². Then substitute into the limit definition.

要通过第一原理求 f(x) = x² 的导数，先写出 f(x + h) = (x + h)²，然后代入极限定义。

f ‘(x) = limₕ→₀ [(x + h)² − x²] / h

Expand the numerator: (x + h)² − x² = x² + 2xh + h² − x² = 2xh + h². Factorise h from the numerator to cancel with the denominator.

展开分子： (x + h)² − x² = x² + 2xh + h² − x² = 2xh + h²。从分子中提取公因式 h，与分母约去。

f ‘(x) = limₕ→₀ [(2xh + h²) / h] = limₕ→₀ (2x + h) = 2x

Therefore the derivative of x² is 2x. This matches the standard power rule and shows how the limit process works in practice.

因此 x² 的导数是 2x。这与标准幂法则一致，也展示了极限过程在实际计算中是如何运作的。

4. Differentiating f(x) = x³ from First Principles | 从第一原理求 f(x) = x³ 的导数

For f(x) = x³, write f(x + h) = (x + h)³. The binomial expansion of (x + h)³ is x³ + 3x²h + 3xh² + h³.

对于 f(x) = x³，写出 f(x + h) = (x + h)³。(x + h)³ 的二项式展开为 x³ + 3x²h + 3xh² + h³。

f ‘(x) = limₕ→₀ [(x + h)³ − x³] / h

Substitute the expansion and simplify: (x + h)³ − x³ = 3x²h + 3xh² + h³. Divide every term by h.

代入展开式并化简： (x + h)³ − x³ = 3x²h + 3xh² + h³。每一项都除以 h。

f ‘(x) = limₕ→₀ (3x² + 3xh + h²) = 3x²

Because 3xh and h² both tend to zero as h tends to zero, the limit is 3x². This method generalises to higher powers of x.

由于当 h 趋于零时，3xh 和 h² 都趋于零，因此极限为 3x²。这种方法可以推广到 x 的更高次幂。

5. The General Power Rule and Its Link | 一般幂法则及其联系

From the results for x² and x³, a clear pattern emerges. For any positive integer power n, the derivative of xⁿ is n xⁿ⁻¹. This is the general power rule.

从 x² 和 x³ 的结果可以归纳出一个清晰规律。对于任意正整数指数 n，xⁿ 的导数是 n xⁿ⁻¹。这就是一般幂法则。

d/dx (xⁿ) = n xⁿ⁻¹

Using the binomial theorem, the expansion of (x + h)ⁿ begins with xⁿ + n xⁿ⁻¹ h. The term xⁿ cancels with the original f(x), and after dividing by h, only n xⁿ⁻¹ remains when h tends to zero. All other terms still contain h and therefore vanish.

利用二项式定理，(x + h)ⁿ 的展开式以 xⁿ + n xⁿ⁻¹ h 开头。xⁿ 与原函数 f(x) 相消，除以 h 后，当 h 趋于零时只剩下 n xⁿ⁻¹。其余所有项仍含有 h，因此都趋于零。

Edexcel exam questions may ask you to prove a derivative from first principles for a specific power, rather than quoting the power rule. Always show the full limiting process.

爱德思考试题可能要求你对某一个具体幂函数用第一原理证明其导数，而不是直接引用幂法则。务必展示完整的极限过程。

6. Differentiating f(x) = √x from First Principles | 从第一原理求 f(x) = √x 的导数

For f(x) = √x, the first-principles quotient contains radicals, so direct expansion is not possible. The standard technique is to multiply the numerator and denominator by the conjugate of the numerator.

对于 f(x) = √x，第一原理的商式中含有根式，因此无法直接展开。标准做法是将分子和分母同时乘以分子的共轭式。

f ‘(x) = limₕ→₀ [√(x + h) − √x] / h

Multiply by [√(x + h) + √x] / [√(x + h) + √x]. The numerator becomes (√(x + h) − √x)(√(x + h) + √x) = (x + h) − x = h.

乘以 [√(x + h) + √x] / [√(x + h) + √x]。分子变为 (√(x + h) − √x)(√(x + h) + √x) = (x + h) − x = h。

f ‘(x) = limₕ→₀ h / [h(√(x + h) + √x)] = limₕ→₀ 1 / (√(x + h) + √x)

As h tends to zero, the denominator tends to √x + √x = 2√x. Therefore the derivative is 1 / (2√x), which is valid for x > 0.

当 h 趋于零时，分母趋于 √x + √x = 2√x。因此导数为 1 / (2√x)，该结果在 x > 0 时成立。

7. Differentiating f(x) = 1/x from First Principles | 从第一原理求 f(x) = 1/x 的导数

Rational functions such as f(x) = 1/x also appear in first-principles questions. Begin with the limit definition and combine the two fractions in the numerator over a common denominator.

像 f(x) = 1/x 这样的有理函数也会出现在第一原理题目中。从极限定义开始，将分子中的两个分数通分合并。

f ‘(x) = limₕ→₀ [1/(x + h) − 1/x] / h

Write 1/(x + h) − 1/x = [x − (x + h)] / [x(x + h)] = −h / [x(x + h)]. Then divide by h.

写出 1/(x + h) − 1/x = [x − (x + h)] / [x(x + h)] = −h / [x(x + h)]。然后除以 h。

f ‘(x) = limₕ→₀ [−h / (h x(x + h))] = limₕ→₀ [−1 / (x(x + h))] = −1 / x²

This result is consistent with the power rule for negative powers: x⁻¹ differentiates to −x⁻², or −1 / x².

这个结果与负指数幂法则一致：x⁻¹ 求导得到 −x⁻²，也就是 −1 / x²。

8. Differentiability and Continuity | 可导性与连续性

A function is differentiable at a point if the first-principles limit exists and gives the same value from both the left and the right. This requires the function to be defined near the point and to have a slope that does not change abruptly.

如果第一原理极限存在，并且从左、右两侧趋近时得到相同的值，那么函数在该点可导。这要求函数在该点附近有定义，且斜率不发生突变。

Differentiability implies continuity, but continuity does not imply differentiability. For example, f(x) = |x| is continuous at x = 0 but is not differentiable there because the left-hand derivative is −1 and the right-hand derivative is 1.

可导性蕴含连续性，但连续性不蕴含可导性。例如 f(x) = |x| 在 x = 0 处连续，但不可导，因为左导数为 −1，右导数为 1。

In first-principles calculations, always check that the final limit exists. If the limit depends on the direction from which h approaches zero, the derivative does not exist at that point.

在第一原理计算中，始终要检查最终极限是否存在。如果极限依赖于 h 从哪个方向趋近于零，那么该点处导数不存在。

9. Sign of the Derivative and Gradient Behaviour | 导数符号与梯度行为

Once f ‘(x) is known, its sign tells you how the original function behaves. If f ‘(x) > 0 on an interval, f(x) is increasing there. If f ‘(x) < 0, f(x) is decreasing. If f '(x) = 0, the graph has a stationary point.

一旦求出 f ‘(x)，它的符号就能告诉你原函数的变化趋势。如果在某个区间内 f ‘(x) > 0，那么 f(x) 在该区间递增；如果 f ‘(x) < 0，则 f(x) 递减；如果 f '(x) = 0，则图像存在驻点。

For example, f(x) = x² has f ‘(x) = 2x. The derivative is negative when x < 0, positive when x > 0, and zero at x = 0. This matches the shape of the parabola with its minimum at the origin.

例如 f(x) = x² 的导数 f ‘(x) = 2x。当 x < 0 时导数为负，当 x > 0 时导数为正，在 x = 0 处导数为零。这与抛物线在原点处取最小值的图像形状一致。

Understanding the sign of the derivative helps you connect first principles to graph sketching and optimisation problems later in the Edexcel specification.

理解导数的符号有助于你将第一原理与爱德思大纲后续的图像绘制和最优化问题联系起来。

10. Edexcel Exam-Style Worked Example | 爱德思考试风格示例

An Edexcel exam question might state: use first principles to show that the derivative of f(x) = x² + 3x is f ‘(x) = 2x + 3. You must not simply use the power rule.

爱德思考试题可能会要求：用第一原理证明 f(x) = x² + 3x 的导数为 f ‘(x) = 2x + 3。你不能直接使用幂法则。

Start with f(x + h) = (x + h)² + 3(x + h) = x² + 2xh + h² + 3x + 3h. Subtract f(x) = x² + 3x.

先写出 f(x + h) = (x + h)² + 3(x + h) = x² + 2xh + h² + 3x + 3h。再减去 f(x) = x² + 3x。

f(x + h) − f(x) = 2xh + h² + 3h

Divide by h: (2xh + h² + 3h) / h = 2x + h + 3. Then take the limit as h tends to zero.

除以 h： (2xh + h² + 3h) / h = 2x + h + 3。然后令 h 趋于零取极限。

f ‘(x) = limₕ→₀ (2x + h + 3) = 2x + 3

Always write the limit symbol at every step before the final evaluation. This shows the examiner that you understand the limiting process, not just the algebra.

在最终求极限之前，每一步都要写出极限符号。这向评分人表明你理解极限过程，而不仅仅是代数运算。

11. Common Mistakes to Avoid | 常见错误与避免方法

One common mistake is forgetting to include the limit symbol limₕ→₀ throughout the working. Without it, the expression [f(x + h) − f(x)] / h is only an average gradient, not a derivative.

一个常见错误是在整个计算过程中忘记写上极限符号 limₕ→₀。没有极限符号，表达式 [f(x + h) − f(x)] / h 只是平均梯度，而不是导数。

Another error is cancelling h incorrectly. Only cancel a factor of h after it has been factorised from every term in the numerator. For example, in (2xh + h² + 3h) / h, every term contains h, so cancellation is valid.

另一个错误是约去 h 时操作不当。只有当分子中的每一项都含有因子 h 时，才能约去 h。例如在 (2xh + h² + 3h) / h 中，每一项都含有 h，因此约分是有效的。

Students also sometimes expand (x + h)² incorrectly as x² + h². The correct expansion is x² + 2xh + h². Similarly, (x + h)³ is not x³ + h³; it must be expanded fully using the binomial theorem or repeated multiplication.

学生有时会错误地把 (x + h)² 展开成 x² + h²。正确的展开是 x² + 2xh + h²。同样地， (x + h)³ 也不是 x³ + h³，而必须使用二项式定理或逐项乘法完全展开。

Finally, do not substitute h = 0 too early. If the denominator becomes zero, the expression is undefined. You must simplify algebraically until the denominator no longer contains h, then take the limit.

最后，不要过早代入 h = 0。如果分母为零，表达式就没有定义。必须先通过代数化简，使分母不再含有 h，然后再取极限。

12. Summary and Key Points | 总结与核心要点

The derivative from first principles is defined by f ‘(x) = limₕ→₀ [f(x + h) − f(x)] / h. This formula is the foundation of all differentiation in Edexcel A-Level Mathematics.

第一原理导数定义为 f ‘(x) = limₕ→₀ [f(x + h) − f(x)] / h。这个公式是爱德思 A-Level 数学所有微分内容的基础。

For polynomial functions, expand the numerator, cancel the common factor h, and then let h tend to zero. For functions involving roots or fractions, use algebraic techniques such as rationalising the numerator or combining fractions before taking the limit.

对于多项式函数，先展开分子，约去公因式 h，再令 h 趋于零。对于含有根式或分式的函数，在取极限之前要使用有理化分子或通分等代数技巧。

Key results to remember include: d/dx (x²) = 2x, d/dx (x³) = 3x², d/dx (√x) = 1/(2√x), and d/dx (1/x) = −1/x². These can all be derived from the same limit definition.

需要牢记的核心结果包括：d/dx (x²) = 2x，d/dx (x³) = 3x²，d/dx (√x) = 1/(2√x)，以及 d/dx (1/x) = −1/x²。这些都可以从同一个极限定义推导出来。

Practise writing clear, logical first-principles proofs. Edexcel questions reward full working, correct limit notation, and correct algebraic simplification. Mastering this topic will strengthen your understanding of tangents, gradients, stationary points, and later applications in mechanics and statistics.

多练习写出清晰、有条理的第一原理证明。爱德思考试重视完整步骤、正确的极限符号和准确的代数化简。掌握这个主题将加深你对切线、梯度、驻点以及后续力学和统计应用的理解。

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Differentiation from First Principles | 第一原理求导