📚 Exercise 20A: Introduction to Differentiation – Power Rule and Sum/Difference Rules | 练习20A：导数入门——幂法则与和差法则
Exercise 20A is often the first set of practice problems that IB Mathematics students encounter when beginning differential calculus. Mastery of the basic derivative rules – especially the power rule and the sum/difference rules – is essential for success across both Analysis and Approaches and Applications and Interpretation courses. This article revisits the foundational concepts, walks through worked examples, and provides strategies for tackling typical Ex. 20A problems with confidence.
练习20A通常是IB数学学生初次接触微分计算时遇到的第一组练习题。掌握基本的导数法则——尤其是幂法则与和差法则——对于在分析与方法、应用与解释两类课程中取得成功至关重要。本文重温基础概念，通过解题示例进行讲解，并提供策略，帮助你有信心地应对典型的练习20A题目。
1. The Derivative as a Rate of Change | 导数作为变化率
n measures how one quantity changes with respect to another. If y = f(x), the derivative f' x. Graphically, it represents the slope of the tangent line to the curve at a given point.
微分衡量的是一个量相对于另一个量的变化情况。如果 y = f(x)，那么导数 f'(x) 给出的是 y 对 x 的瞬时变化率。从图像上看，它代表曲线上某一点处切线的斜率。
The formal definition uses limits: f' le you do not need to compute every derivative by this limit definition after mastering the short‑cut rules, understanding it is crucial for interpreting results and tackling proof‑based questions.
正式定义使用极限：f'(x) = lim₍ₕ→₀₎ [f(x+h) - f(x)] / h。虽然在掌握了快捷法则后并不需要每次都通过这个极限定义来计算导数，但理解它对于解释结果和解决证明类题目至关重要。
2. Derivative as a Function vs. Derivative at a Point | 导函数与某点处的导数
When we differentiate f(x), we obtain a new function f' can be evaluated at any x in the domain where the function is differentiable. The value f'(a) is the gradient of the tangent at x = a.
当我们对 f(x) 求导时，得到一个新的函数 f'(x)，它可以在函数可导的任意 x 处求值。数值 f'(a) 就是曲线在 x = a 处的切线斜率。
In IB questions, you may be asked to find f'(x) first, and then substitute a specific value. This two‑step process is common in exercises like Ex. 20A, where you practise applying rules to generate derivative functions.
在IB考题中，你可能需要先求出 f'(x)，然后再代入具体的数值。这种两步过程在类似练习20A的题目中很常见，通过练习你可以学会应用法则来生成导函数。
3. The Power Rule: d/dx [xⁿ] = n xⁿ⁻¹ | 幂法则：d/dx [xⁿ] = n xⁿ⁻¹
differentiate monomials quickly without limits. For example, if y = x⁵, then dy/dx = 5x⁴.
对于任意实数 n，xⁿ 的导数是 n xⁿ⁻¹。这条法则让你无需借助极限就能快速对单项式求导。例如，如果 y = x⁵，那么 dy/dx = 5x⁴。
The power rule works for positive integers, negative integers, fractions, and even irrational exponents, as long as you write the function in the form xⁿ before differentiating. Ex. 20A often includes terms like 1/x² or √x, which must be rewritten as x⁻² or x½ respectively.
幂法则适用于正整数、负整数、分数甚至无理指数，只要你将函数写成 xⁿ 的形式后再求导即可。练习20A中常常包含像 1/x² 或 √x 这样的项，它们需要分别重写为 x⁻² 或 x½。
4. Constant Multiple and Sum/Difference Rules | 常数倍数法则与和差法则
The constant multiple rule states that d/dx [c ⋅ f(x)] = c ⋅ f' r any constant c. The sum rule allows differentiation term by term: d/dx [f(x) ± g(x)] = f'(x) ± g'(x).
常数倍数法则指出，对于任意常数 c，有 d/dx [c ⋅ f(x)] = c ⋅ f'(x)。和差法则允许逐项求导：d/dx [f(x) ± g(x)] = f'(x) ± g'(x)。
e addition/subtraction signs. For y = 3x⁴ - 2x² + 7x - 5, the derivative is 12x³ - 4x + 7.
结合幂法则，这些法则使多项式求导变得非常简单。只需将每个系数乘以指数，指数减1，并保留加减号即可。对于 y = 3x⁴ - 2x² + 7x - 5，导数为 12x³ - 4x + 7。
5. Rewriting Before Differentiating | 求导前先重写表达式
Many functions in Ex. 20A are not initially presented as a sum of xⁿ terms. Expressions like (x+1)/x² or √x + 1/∛x must be simplified algebraically before applying rules. Convert radicals to fractional exponents and split fractions into separate terms.
练习20A中的许多函数最初并不是以 xⁿ 项的和的形式给出的。像 (x+1)/x² 或 √x + 1/∛x 这样的表达式，在应用法则之前必须先进行代数化简。将根式转化为分数指数，并将分式拆分成单独的项。
x⁻¹ + x⁻². Then differentiate to get -x⁻² - 2x⁻³. This rewriting step is frequently tested and is a major source of errors if skipped.
例如，(x+1)/x² = x/x² + 1/x² = x⁻¹ + x⁻²。然后求导得到 -x⁻² - 2x⁻³。这个重写步骤经常是考查重点，如果跳过，很容易出错。
6. Working with Negative and Fractional Exponents | 负指数与分数指数的处理
with a positive exponent in the denominator: -3/x⁴. Similarly, for y = x⅔, dy/dx = (⅔)x⁻⅓.
幂法则可以无缝处理负指数。对于 y = x⁻³，dy/dx = -3x⁻⁴。结果通常写成分母带正指数的形式：-3/x⁴。类似地，对于 y = x⅔，dy/dx = (⅔)x⁻⅓。
IB mark schemes often expect final answers to be given with positive exponents or in rationalised form, depending on the chapter. Check your teacher's preference and make a habit of simplifying derivative expressions fully.
IB的评分方案通常要求最终答案以正指数或有理化形式给出，具体取决于章节。了解老师的偏好，并养成将导数表达式完全化简的习惯。
7. Higher‑Order Derivatives | 高阶导数
The second derivative, written as f''(x) or d²y/dx², is the derivative of the first derivative. It measures the rate of change of the gradient and is used to determine concavity and acceleration. You find it by differentiating f'(x) using the same rules.
二阶导数，写作 f''(x) 或 d²y/dx²，是一阶导数的导数。它衡量斜率的变化率，用于判断曲线的凹凸性和加速度。你可以通过用相同法则对 f'(x) 求导来得到它。
If f'(x) = 6x² - 8x + 3, then f''(x) = 12x - 8. Ex. 20A may include simple higher‑ idea that differentiation can be repeated as long as the function remains differentiable.
如果 f'(x) = 6x² - 8x + 3，那么 f''(x) = 12x - 8。练习20A可能会包含简单的高阶导数题，以强化这样一个观念：只要函数仍可导，求导就可以重复进行。
8. Tangent and Normal Lines – First Applications | 切线与法线——最初的应用
One of the most common follow‑up tasks after finding a derivative is to write the equation of a tangent or normal at a specified point. The gradient mₜ of the tangent is simply f'(a). The normal gradient mₙ is -1 / mₜ (provided mₜ ≠ 0).
在求导之后，最常见的后续任务之一是写出在某指定点处的切线或法线方程。切线的斜率 mₜ 就是 f'(a)。法线的斜率 mₙ 为 -1 / mₜ（只要 mₜ ≠ 0）。
Using the point‑ ckly construct the required linear equation. These problems bridge differentiation rules with coordinate geometry and appear frequently at the end of Ex. 20A sets.
使用点斜式 y - y₁ = m(x - x₁)，你可以快速构造出所需的线性方程。这类问题将求导法则与解析几何联系起来，经常出现在练习20A题组的末尾。
9. Common Mistakes and How to Avoid Them | 常见错误及如何避免
ut multiplying are typical slips. For y = x⁴, the derivative is 4x³, not x³ or 4x⁴. Double‑check each term after differentiating.
忘记乘以原来的指数，或者只改了指数而没有做乘法，这些都是典型的失误。对于 y = x⁴，导数是 4x³，而不是 x³ 或 4x⁴。求导后要仔细检查每一项。
Another error is mishandling constants: the derivative of a constant term like +5 is zero, never 5. Also, when differentiating a sum like 2x² + 4, the constant 4 disappears, but the 2 stays as a coefficient.
另一个错误是常数处理不当：像 +5 这样的常数项的导数为零，绝不是5。此外，当对 2x² + 4 这样的和式求导时，常数4会消失，但2仍然作为系数保留下来。
10. Worked Examples from a Typical Ex. 20A | 典型练习20A解题示例
Example 1: Differentiate f(x) = 5x³ - 2x² + x - 7.
f'(x) = 15x² - 4x + 1.
示例1：对 f(x) = 5x³ - 2x² + x - 7 求导。
f'(x) = 15x² - 4x + 1。
Example 2: Find dy/dx if y = 4/x² + √x.
Rewrite: y = 4x⁻² + x½. Then dy/dx = -8x⁻³ + ½ x⁻½ = -8/x³ + 1/(2√x).
示例2：若 y = 4/x² + √x，求 dy/dx。
重写：y = 4x⁻² + x½。则 dy/dx = -8x⁻³ + ½ x⁻½ = -8/x³ + 1/(2√x)。
Example 3: Determine f''(x) for f(x) = x⁴ - x³.
f'(x) = 4x³ - 3x²; f''(x) = 12x² - 6x.
示例3：对于 f(x) = x⁴ - x³，求 f''(x)。
f'(x) = 4x³ - 3x²；f''(x) = 12x² - 6x。
11. Connecting to IB Exam Questions | 与IB考试题型的衔接
Exam questions may embed differentiation in context: velocity from displacement, marginal cost in economics, or optimisation of an area. Even at this early stage, recognising that the derivative is a tool for finding rates of change helps you interpret word problems.
考试题可能会将求导嵌入具体情境中：从位移求速度、经济学中的边际成本，或者面积的优化问题。即使在这个早期阶段，认识到导数是一种寻找变化率的工具，也能帮助你理解应用题。
Always state the derivative formula clearly, show the algebraic steps, and present the final answer in the simplest form. The discipline developed while solving Ex. 20A builds the automaticity required for Paper 1 non‑calculator sections.
务必清晰地写出导数公式，展示代数步骤，并将最终答案以最简形式呈现。在解决练习20A的过程中养成的严谨习惯，将为试卷一中的非计算器部分打下自动化的基础。
12. Summary and Next Steps | 总结与后续学习步骤
Exercise 20A consolidates the power rule, constant multiple, and sum/difference rules. Through deliberate practice, you transform the process of differentiating polynomials, rational powers, and rewritten expressions into a fluent algebraic skill.
练习20A巩固了幂法则、常数倍数法则以及和差法则。通过有意识的练习，你将把对多项式、有理指数幂以及重写式的求导过程转化为一种娴熟的代数技能。
Once confident with these basics, you are ready to extend the toolkit with the product rule, quotient rule, and chain rule, which will open up differentiation of trigonometric, exponential, and logarithmic functions later in the course.
一旦对这些基础有了信心，你就可以准备好用乘积法则、商法则和链式法则来扩展工具包，这些法则将在课程的后续部分中打开对三角函数、指数函数和对数函数求导的大门。
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