📚 Exercise 10A: Differentiation from First Principles | 练习10A：从第一性原理求导
In IB Mathematics, Exercise 10A typically introduces one of the most foundational ideas in calculus – differentiation from first principles. This approach builds a rigorous understanding of what a derivative truly represents: the instantaneous rate of change of a function, defined as a limit. Mastering this exercise not only prepares you for the rules of differentiation but also deepens your appreciation of how mathematics models change in the real world. In this article, we break down the key concepts, step-by-step techniques, and common pitfalls, followed by practice questions that mirror those you will encounter in Exercise 10A.
在 IB 数学课程中，练习10A通常介绍微积分最核心的思想之一——从第一性原理求导。这种方法可以帮助你严谨地理解导数究竟代表什么：函数在某一点的瞬时变化率，通过极限来定义。掌握这一内容不仅为后续的求导法则打下坚实基础，也能让你更深刻地体会数学如何为现实世界中的变化建模。本文将详细拆解关键概念、逐步推导技巧以及常见错误，最后提供与练习10A类似的典型习题。
1. Understanding the Gradient of a Curve | 理解曲线的梯度
(x), we start by drawing a chord connecting that point to a nearby point. As the second point moves closer, the chord becomes a better approximation of the tangent line at the point of interest.
在坐标几何中，直线的斜率是常数，可以通过两点轻松求得。但对于曲线，陡峭程度在每一点都在变化。要求曲线 y = f(x) 上某一点的斜率，我们首先画一条连接该点和邻近点的弦。当邻近点向定点无限靠近时，这条弦就越来越逼近该点处的切线。
The gradient of the chord is given by the difference quotient (f(x+h) - f(x)) / h, where h is the small horizontal displacement. The true gradient of the curve at x is obtained by making h approach zero, which leads to the concept of a limit. This idea is the heart of differentiation from first principles.
弦的斜率由差商 (f(x+h) - f(x)) / h 表示，其中 h 是水平方向上的微小增量。曲线上 x 处的真实梯度就是当 h 趋近于零时的极限值，从而引出了极限的概念。这一思想正是从第一性原理求导的核心。
2. The Concept of a Limit | 极限的概念
(x)) / h. It is essential to recognise that h never actually equals zero; we examine the behaviour as h gets arbitrarily small.
极限描述的是当输入趋近于某个值时，函数值所趋近的那个数值。对于求导来说，我们关注的是当 h → 0 时差商的极限。用符号可以写成 lim (h → 0) (f(x+h) - f(x)) / h。必须明确，h 永远不会真正等于零；我们考察的是当 h 任意变小时该表达式的趋势。
he meaningless expression 0/0. Instead, algebraic manipulation is used to cancel h before letting h tend to zero. This step is vital in first principles exercises.
理解极限能帮助你避免直接代入 h = 0 的常见错误，那样会得到无意义的 0/0 形式。正确做法是先通过代数变形消去 h，再让 h 趋向于零。这一步在第一性原理的练习中至关重要。
3. Definition of the Derivative | 导数的定义
The derivative of a function f at a point x is defined as f' rovided this limit exists. If the limit exists, we say f is differentiable at x. The derivative f' e to the curve at that point, and it also represents the instantaneous rate of change of f with respect to x.
函数 f 在点 x 处的导数定义为 f'(x) = lim (h → 0) [(f(x+h) - f(x))/h]，只要该极限存在。如果极限存在，我们就说 f 在 x 处可导。导数 f'(x) 表示曲线在该点处切线的斜率，同时也代表了 f 关于 x 的瞬时变化率。
This definition is what we call 'differentiation from first principles'. In many textbooks, Exercise 10A asks you to apply this definition to various simple functions before learning the shortcut rules such as the power rule, product rule, and chain rule.
这个定义就是我们所说的“从第一性原理求导”。在许多教材中，练习10A要求学生在学习幂法则、乘法法则和链式法则等快捷方法之前，先用这一定义求几种简单函数的导数。
4. First Principles Formula | 第一性原理公式
f'(x) = lim (h → 0) [f(x+h) - f(x)] / h
d then cancel h wherever possible. Only after cancellation do you take the limit as h → 0.
这个紧凑的公式是练习10A的起点。对于给定的具体函数 f(x)，你把 f(x+h) 替换为函数在 (x+h) 处的取值，将分子展开并化简，然后尽可能约去 h。只有在约去 h 之后，才令 h 趋近于零取极限。
This process must be followed precisely; any attempt to evaluate the limit before cancelling h will lead to an indeterminate form. The first principles approach ensures you see how the derivative emerges from the geometry of the curve and the algebra of limits.
这个过程必须严格遵守；任何在约去 h 之前就取极限的尝试都会导致不定式。第一性原理的方法能让你清楚地看到导数如何从曲线的几何特征和极限的代数运算中产生。
5. Step-by-Step Process | 逐步求导过程
To differentiate f(x) from first principles, follow these four essential steps:
从第一性原理求 f(x) 的导数，遵循以下四个关键步骤：
- Step 1: Write down f(x+h) by replacing every x in f(x) with (x+h). / 将 f(x) 中的每一个 x 替换为 (x+h)，写出 f(x+h)。
- Step 2: Form the difference quotient: [f(x+h) - f(x)] / h. / 写出差商：[f(x+h) - f(x)] / h。
- Step 3: Simplify the numerator algebraically and factor out h if possible. / 用代数方法化简分子，如果可能就提取因子 h。
- Step 4: Cancel h from the numerator and denominator, then take the limit as h → 0 to find f'(x). / 约去分子和分母中的 h，然后取 h → 0 时的极限，得到 f'(x)。
Presenting your work clearly with these steps will earn full marks in IB examinations. Even if a short-cut rule exists, exam questions may specify 'use first principles', so you must show the limit process.
在 IB 考试中，清晰地展示这些步骤将帮助你获得满分。即使存在快捷求导公式，考题也可能明确要求“使用第一性原理”，因此你必须写出极限过程。
6. Example: Differentiating f(x)=x² | 示例：求 f(x)=x² 的导数
Let f(x) = x². Then f(x+h) = (x+h)² = x² + 2xh + h². The difference quotient becomes:
设 f(x) = x²，则 f(x+h) = (x+h)² = x² + 2xh + h²。差商为：
[f(x+h) - f(x)] / h = (x² + 2xh + h² - x²) / h = (2xh + h²) / h
h (since h ≠ 0 before taking the limit), giving 2x + h. Finally, take the limit as h → 0:
从分子中提取 h：h(2x + h) / h。约去 h（因为在取极限前 h ≠ 0），得到 2x + h。最后，取 h → 0 时的极限：
f'(x) = lim (h → 0) (2x + h) = 2x
This shows that the derivative of x² is 2x, a result you will later recognise as a case of the power rule. The first principles derivation confirms why the rule works.
这表明 x² 的导数是 2x，之后你会认识到这是幂法则的一个特例。通过第一性原理的推导，你能够理解为什么这条法则成立。
7. Example: Differentiating f(x)=x³ | 示例：求 f(x)=x³ 的导数
For f(x) = x³, we have f(x+h) = (x+h)³ = x³ + 3x²h + 3xh² + h³. Subtracting f(x) gives:
对于 f(x) = x³，有 f(x+h) = (x+h)³ = x³ + 3x²h + 3xh² + h³。减去 f(x) 得到：
f(x+h) - f(x) = 3x²h + 3xh² + h³
Divide by h: (3x²h + 3xh² + h³) / h = 3x² + 3xh + h². Take the limit h → 0:
除以 h：(3x²h + 3xh² + h³) / h = 3x² + 3xh + h²。取极限 h → 0：
f'(x) = lim (h → 0) (3x² + 3xh + h²) = 3x²
Again, the first principles approach yields f' expanding cubes or higher powers becomes routine, and you will notice a pattern emerging.
再次，第一性原理方法给出 f'(x) = 3x²。经过练习，展开三次方或更高次方将变得常规，你也会注意到一种规律正在显现。
8. Differentiating f(x)=c (Constant) | 求常数函数的导数
x+h) = c, and the difference quotient is (c - c) / h = 0 / h = 0. Taking the limit gives f'(x) = 0. This makes intuitive sense: a horizontal line has zero gradient everywhere.
一个重要的特例是求常数函数 f(x) = c 的导数，其中 c 是实数。此时 f(x+h) = c，差商为 (c - c) / h = 0 / h = 0。取极限得到 f'(x) = 0。这在直观上也合理：水平直线的斜率处处为零。
In Exercise 10A, you may be asked to differentiate a constant from first principles to reinforce the concept that the derivative represents rate of change. If a quantity is not changing, its rate of change is zero.
在练习10A中，可能会要求你用第一性原理求常数的导数，以强化导数代表变化率的概念。如果一个量不发生变化，它的变化率就是零。
9. Differentiating f(x)=axⁿ | 幂函数求导
ing the binomial theorem to expand (x+h)ⁿ, the numerator becomes a[(x+h)ⁿ - xⁿ]. The term xⁿ cancels, and the smallest power of h in the remaining expression is h¹, allowing cancellation with the denominator. After taking the limit, we obtain:
考虑一般幂函数 f(x) = axⁿ，其中 n 为正整数。利用二项式定理展开 (x+h)ⁿ，分子变为 a[(x+h)ⁿ - xⁿ]。xⁿ 项被消去，剩余表达式中 h 的最低次幂为一次，因此可与分母约去。取极限后得到：
f'(x) = a n xⁿ⁻¹
This is the power rule, proven from first principles. In Exercise 10A, you may not be required to prove the general case, but applying this to specific powers like x⁴ or 5x³ is typical.
这就是由第一性原理证明的幂法则。在练习10A中，你可能不需要证明一般情形，但将其应用到具体幂次（如 x⁴ 或 5x³）是典型的练习内容。
10. Generalising the Power Rule | 幂法则的推广
Although first principles can be used for any positive integer exponent, the power rule f' y real number n. In the IB course, you will eventually differentiate functions like x⁻², x1/2, and xπ using this rule. However, Exercise 10A usually sticks to simple polynomials where the first principles algebra is manageable.
尽管第一性原理可用于任何正整数指数，幂法则 f'(x) = n xⁿ⁻¹ 实际上对所有实数 n 都成立。在 IB 课程中，你最终会使用该法则求形如 x⁻²、x1/2 甚至 xπ 的导数。但练习10A通常只涉及简单的多项式，因为此时的代数运算较为可控。
Understanding the derivation for natural number exponents gives you the confidence that the rule is not arbitrary, and it prepares you for more advanced topics such as implicit differentiation and differential equations.
理解正整数指数的推导过程能让你确信幂法则并非凭空产生，也为你学习隐函数求导和微分方程等更高阶的主题做好准备。
11. Common Mistakes to Avoid | 常见错误避免
f(x+h) - f(x) = (x+h)² - x² = x² + h² - x², missing the 2xh term. Always expand carefully.
最常见的错误之一就是没有正确减去 f(x)，尤其是涉及负号时。例如对于 f(x) = x²，有些学生会错误地写成 f(x+h) - f(x) = (x+h)² - x² = x² + h² - x²，遗漏了 2xh 项。务必仔细展开。
back and check your algebra. Also, do not set h = 0 before cancelling, as this gives 0/0.
另一个错误是当分子无法提取因子 h 时强行约分。请记住，必须先确保分子中的每一项都含有因子 h。如果没有，回头检查代数运算。此外，不要在约去 h 之前就让 h = 0，否则会得到 0/0。
Finally, many students forget to take the limit at the end, leaving an expression containing h. The final answer for the derivative must not include h.
最后，许多学生会忘记在最后一步取极限，从而在结果中留下含 h 的表达式。导数作为最终答案，绝对不能含有 h。
12. Practice Questions from Exercise 10A | 练习10A 的练习题目
Below are typical problems you would encounter in Exercise 10A. Use first principles for each, showing full working.
以下是你会在练习10A中遇到的典型题目。对每一个问题使用第一性原理，并展示完整的运算过程。
- 1. Differentiate f(x) = x² - 5x from first principles. / 从第一性原理求 f(x) = x² - 5x 的导数。
- 2. Find the derivative of g(x) = 2x³ + 3x. / 求 g(x) = 2x³ + 3x 的导数。
- 3. Determine f'(x) for f(x) = 7. / 求 f(x) = 7 的导数 f'(x)。
- 4. Use first principles to differentiate h(x) = 4x² - x³. / 用第一性原理求 h(x) = 4x² - x³ 的导数。
- tive of f(x) = 1/x is -1/x² using the limit definition. (This is a challenge extension.) / 利用极限定义证明 f(x) = 1/x 的导数为 -1/x²。（挑战性拓展）
Attempt these after reviewing the step-by-step examples. Drawing a tangent graph alongside your algebraic work can help connect the numerical process to the geometric meaning of the derivative.
在复习完逐步示例后尝试这些习题。在用代数方法计算的同时，画出切线草图，有助于将数值过程与导数的几何意义联系起来。
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