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 uses the fundamental definition of the derivative as a limit to find the gradient of a curve at any point. This method, rooted in the concept of instantaneous rate of change, lies at the heart of calculus and underpins all differentiation rules used at IB level. Mastering this approach not only builds deep understanding but also sharpens algebraic skills essential for higher‑level mathematics.

从第一原理求导利用导数作为极限的基本定义来计算曲线上任意一点的梯度。这种方法基于瞬时变化率的概念，是微积分的核心，支撑着IB阶段所有的求导法则。掌握这种方法不仅能加深理解，还能锻炼高等数学必备的代数技能。

---

1. The Idea of a Limit | 极限的概念

To understand differentiation from first principles, we first need a solid grasp of a limit. Informally, the limit of a function f(x) as x approaches a value c is the number that f(x) gets arbitrarily close to, provided the values of x stay near c. This idea allows us to examine behaviour at a point without requiring the function to be defined there.

要理解从第一原理求导，首先需要牢固掌握极限的概念。通俗地说，函数f(x)当x趋于某个值c时的极限，是指只要x的值保持在c附近，f(x)就可以无限接近的那个数。这个概念使我们能研究函数在某一点的行为，而不要求函数在该点本身有定义。

In differentiation, the key limit is the difference quotient [f(x+h) – f(x)] / h as h approaches 0. Here h represents a tiny change in x, and the fraction gives the slope of a secant line joining two close points on the curve. The limit captures the slope of the tangent when those two points practically coincide.

在求导中，关键的极限是当h趋近于0时差商[f(x+h) – f(x)] / h的极限。这里h代表x的一个微小变化，这个分式给出了曲线上两个邻近点连接的割线斜率。当这两个点几乎重合时，极限就捕捉到了切线的斜率。

---

2. The Definition of the Derivative | 导数的定义

For a function y = f(x), the derivative at a general point x is defined by

对于函数y = f(x)，其在一般点x处的导数定义为

f ‘(x) = lim (h → 0) [f(x + h) – f(x)] / h

This expression is often read as “f prime of x equals the limit as h goes to zero of f of x plus h minus f of x, all over h”. The derivative f ‘(x) itself is a new function that gives the gradient of the tangent to the curve y = f(x) at any x for which the limit exists.

这个表达式常读作“f撇x等于当h趋于0时，f(x加h)减f(x)的差除以h的极限”。导数f ‘(x)本身是一个新函数，给出曲线y = f(x)在任意使该极限存在的x处的切线斜率。

If the limit exists at a particular x, we say f is differentiable at that point. The limit must be the same whether h approaches 0 from the positive side (h → 0⁺) or the negative side (h → 0⁻). For polynomial, root, trigonometric and exponential functions, this condition holds over their natural domains.

如果极限在特定x处存在，我们就说f在该点可导。无论h从正侧(h → 0⁺)还是负侧(h → 0⁻)趋于0，极限都必须是相同的。对于多项式、根式、三角函数和指数函数，这个条件在其自然定义域内都是成立的。

---

3. Notation for Derivatives | 导数的记法

Several notations are used for the derivative of y = f(x). The most common in IB are:

对于y = f(x)的导数，有多种记法。IB中最常用的有：

• f ‘(x) – Lagrange notation, read as “f prime of x”.
• dy/dx – Leibniz notation, emphasising the rate of change of y with respect to x.
• df/dx – similar to dy/dx but using the function name.

• f ‘(x)——拉格朗日记法，读作“f撇x”。
• dy/dx——莱布尼茨记法，强调y关于x的变化率。
• df/dx——与dy/dx类似，但使用了函数名。

All these notations represent the same concept: the limit of the difference quotient. In first‑principles work, we often start with f ‘(x) and later switch to dy/dx when solving differential equations or related rates.

所有这些记法都表示同一个概念：差商的极限。在第一原理的推演中，我们常从f ‘(x)入手，而在解微分方程或相关变化率问题时则转换为dy/dx。

---

4. Step‑by‑Step Process | 逐步求解方法

Applying the first‑principles definition systematically makes the algebra manageable. Follow these steps:

系统地运用第一原理定义能使代数推导变得可控。请遵循以下步骤：

Step 1: Write down f(x) and form f(x + h) by replacing every x with (x + h).
第1步：写下f(x)，将每个x替换为(x + h)得到f(x + h)。

Step 2: Construct the difference quotient [f(x + h) – f(x)] / h.
第2步：构建差商[f(x + h) – f(x)] / h。

Step 3: Simplify the numerator by expanding, factorising, or rationalising as needed. The goal is to cancel h in the denominator.
第3步：通过展开、因式分解或有理化化简分子。目标是约去分母中的h。

Step 4: Cancel any common factor of h.
第4步：约去任何公因式h。

Step 5: Take the limit as h → 0. Any remaining term that contains h will vanish, leaving the derivative f ‘(x).
第5步：取h → 0时的极限。任何含h的项都将消失，得到导数f ‘(x)。

This procedure can be applied to polynomials, rational functions, root functions, and eventually to trig functions using compound angle identities.

该步骤可应用于多项式、有理函数、根式函数，最终还可结合复合角公式应用于三角函数。

---

5. Example: f(x) = x² | 示例：f(x) = x²

Let f(x) = x². We work through each step:

令f(x) = x²。我们逐步推演：

Step 1: f(x + h) = (x + h)² = x² + 2xh + h².
第1步：f(x + h) = (x + h)² = x² + 2xh + h²。

Step 2: f(x + h) – f(x) = (x² + 2xh + h²) – x² = 2xh + h².
第2步：f(x + h) – f(x) = (x² + 2xh + h²) – x² = 2xh + h²。

Step 3: Difference quotient = (2xh + h²) / h.
第3步：差商 = (2xh + h²) / h。

Step 4: Factor h: (h(2x + h)) / h = 2x + h, provided h ≠ 0.
第4步：提取公因式h：(h(2x + h)) / h = 2x + h，假设h ≠ 0。

Step 5: Take the limit: f ‘(x) = lim (h → 0) (2x + h) = 2x.
第5步：取极限：f ‘(x) = lim (h → 0) (2x + h) = 2x。

Thus the derivative of x² is 2x. This matches the power rule and confirms the first‑principles method.

因此x²的导数是2x。这与幂法则一致，验证了第一原理方法。

---

6. Example: f(x) = x³ | 示例：f(x) = x³

Now consider f(x) = x³. The binomial expansion is needed: (x + h)³ = x³ + 3x²h + 3xh² + h³.
现在考虑f(x) = x³。需要使用二项式展开：(x + h)³ = x³ + 3x²h + 3xh² + h³。

Then f(x + h) – f(x) = (x³ + 3x²h + 3xh² + h³) – x³ = 3x²h + 3xh² + h³.
那么f(x + h) – f(x) = (x³ + 3x²h + 3xh² + h³) – x³ = 3x²h + 3xh² + h³。

Difference quotient = (3x²h + 3xh² + h³) / h = 3x² + 3xh + h² (after cancelling h).
差商 = (3x²h + 3xh² + h³) / h = 3x² + 3xh + h²（约去h后）。

Taking the limit h → 0 gives f ‘(x) = 3x² + 3x·0 + 0² = 3x².
取极限h → 0，得f ‘(x) = 3x² + 3x·0 + 0² = 3x²。

Again, the first‑principles result agrees with the power rule: d/dx (x³) = 3x². The process highlights how terms containing h vanish, leaving only those that originally had a single factor of h.

同样，第一原理的结果与幂法则一致：d/dx (x³) = 3x²。这个过程突显了含h的项是如何消失的，只留下原来带有一个h因子的项。

---

7. Example: f(x) = √x | 示例：f(x) = √x

For f(x) = √x (x ≥ 0), rationalising the numerator is the key technique. Form f(x + h) = √(x + h).
对于f(x) = √x（x ≥ 0），分子有理化是关键技巧。构造f(x + h) = √(x + h)。

Difference quotient: [√(x + h) – √x] / h.
差商：[√(x + h) – √x] / h。

Multiply numerator and denominator by the conjugate: (√(x + h) + √x). This gives
分子分母同乘共轭式：(√(x + h) + √x)。得到

[(x + h) – x] / [h(√(x + h) + √x)] = h / [h(√(x + h) + √x)]

Cancel h (h ≠ 0): we are left with 1 / (√(x + h) + √x).
约去h（h ≠ 0）：得到1 / (√(x + h) + √x)。

Take the limit h → 0: f ‘(x) = 1 / (√(x + 0) + √x) = 1 / (2√x).
取极限h → 0：f ‘(x) = 1 / (√(x + 0) + √x) = 1 / (2√x)。

Thus d/dx (√x) = 1/(2√x), which is exactly what the power rule gives if we write √x = x^(1/2) and differentiate: (1/2)x^(–1/2) = 1/(2√x).
因此d/dx (√x) = 1/(2√x)，这正好是把√x写作x^(1/2)后用幂法则求导的结果：(1/2)x^(–1/2) = 1/(2√x)。

---

8. Generalising to the Power Rule | 推广到幂法则

From the first‑principles examples, we see a pattern: d/dx (x) = 1x⁰, d/dx (x²) = 2x¹, d/dx (x³) = 3x². Using the binomial theorem for (x + h)^n, the general result emerges:
从这些第一原理的例子中，我们可以看到一种模式：d/dx (x) = 1x⁰, d/dx (x²) = 2x¹, d/dx (x³) = 3x²。对于(x + h)^n运用二项式定理，可得一般结果：

d/dx (x^n) = n x^(n–1) for any real constant n

This is the power rule. For integer n, the proof uses the binomial expansion, where the x^(n–1)h term yields the derivative after cancelling h. For non‑integer n, the proof is more subtle but the rule still holds.

这就是幂法则。对于整数n，证明使用二项式展开，其中x^(n–1)h项在约去h后给出导数。对于非整数n，证明更为精细，但法则仍然成立。

IB students are expected to know how to prove the power rule for positive integer exponents using first principles, as it reinforces both algebraic manipulation and the limit concept.

IB学生需掌握如何使用第一原理证明正整数指数的幂法则，因为这同时强化了代数操作和极限概念。

---

9. Constant and Linear Functions | 常数函数与线性函数

For a constant function f(x) = c, f(x + h) = c. The difference quotient becomes (c – c)/h = 0/h = 0. Hence f ‘(x) = 0. This makes intuitive sense: a horizontal line has zero gradient.
对于常数函数f(x) = c，f(x + h) = c。差商变为(c – c)/h = 0/h = 0。因此f ‘(x) = 0。这很直观：水平线的梯度为零。

For f(x) = ax + b (a linear function), f(x + h) = a(x + h) + b = ax + ah + b. Then f(x + h) – f(x) = ah. The difference quotient is ah/h = a. Taking the limit gives f ‘(x) = a. The derivative of any linear function is simply its slope.
对于f(x) = ax + b（线性函数），f(x + h) = a(x + h) + b = ax + ah + b。那么f(x + h) – f(x) = ah。差商为ah/h = a。取极限得到f ‘(x) = a。任何线性函数的导数就是它的斜率。

---

10. Common Mistakes and Tips | 常见错误与提示

When differentiating from first principles, students often stumble on algebra. Here are pitfalls to avoid:
用第一原理求导时，学生常在代数上出错。以下是一些要避免的陷阱：

• Forgetting to cancel h properly: Always factor h from the numerator before cancelling. Only terms with a factor h will survive the limit.
  未能正确约去h：约分前务必从分子中提取公因式h。只有含有h因子的项才会在极限下保留。
• Misapplying the binomial expansion: For (x + h)³, missing the 3xh² term leads to an incorrect derivative. Write each term carefully.
  二项式展开出错：对于(x + h)³，漏掉3xh²项会导致导数错误。请仔细写出每一项。
• Rationalising incorrectly: With √(x + h) – √x, multiply by the conjugate and simplify fully before taking the limit.
  有理化不当：处理√(x + h) – √x时，在取极限前要乘共轭式并彻底化简。
• Using the definition at a specific point: If asked for f ‘(a), set up [f(a + h) – f(a)]/h and let h → 0. Do not differentiate the whole function first unless allowed.
  求特定点处的导数：如果要求f ‘(a)，就建立[f(a + h) – f(a)]/h并令h → 0。除非题目允许，不要先对整个函数求导再代入。

Practice each function type multiple times. The more fluent you become with the algebra, the easier it is to tackle exam questions involving limits, continuity and differentiability.

每种函数类型都要反复练习。你对代数越熟练，就越容易应对涉及极限、连续性和可导性的考试题目。

---

11. Beyond Polynomials: Trig and Exponentials | 超越多项式：三角函数与指数函数

While IB may not always require a full first‑principles proof for sin x, eˣ or ln x, understanding the structure is valuable. For f(x) = sin x, the difference quotient uses the identity sin(x + h) = sin x cos h + cos x sin h. The limit then involves the known results lim(h→0) (sin h)/h = 1 and lim(h→0) (cos h – 1)/h = 0, leading to f ‘(x) = cos x.
尽管IB不总要求对sin x、eˣ或ln x进行完整的第一原理证明，但理解其结构很有价值。对于f(x) = sin x，差商使用恒等式sin(x + h) = sin x cos h + cos x sin h。极限随后用到已知结果lim(h→0) (sin h)/h = 1和lim(h→0) (cos h – 1)/h = 0，进而得出f ‘(x) = cos x。

For f(x) = eˣ, the difference quotient involves eˣ(e^h – 1)/h. The special limit lim(h→0) (e^h – 1)/h = 1 yields f ‘(x) = eˣ. These examples illustrate that first principles is the foundation for every derivative formula we use.

对于f(x) = eˣ，差商包含eˣ(e^h – 1)/h。特殊极限lim(h→0) (e^h – 1)/h = 1给出f ‘(x) = eˣ。这些例子说明，第一原理是我们使用的每一个求导公式的基础。

---

12. Why First Principles Matter in IB | 第一原理在IB中的重要性

The IB syllabus explicitly includes differentiation from first principles, particularly in the Analysis & Approaches (AA) course. Questions may ask students to find the derivative of a simple polynomial, root or rational function using the limit definition. Additionally, the concept underpins the formal definition of the derivative, which is tested in Paper 1 with no calculator.

IB大纲明确包含从第一原理求导，特别是在分析与方法(AA)课程中。考题可能要求学生用极限定义求出简单多项式、根式或有理函数的导数。此外，该概念支撑着导数的正式定义，这在不可使用计算器的试卷一中有考查。

Mastering first principles also strengthens the ability to prove the sum, product and chain rules. It connects the graphical idea of tangent slope with algebraic manipulation, bridging the gap between intuitive understanding and rigorous mathematics.

掌握第一原理还能增强证明和法则、积法则和链式法则的能力。它把切线斜率的图形概念与代数操作连接起来，弥合了直观理解与严谨数学之间的鸿沟。

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）