📚 Exercise 23B.1: Differentiation of Polynomial Functions | 练习题23B.1：多项函数的微分
Welcome to this detailed walkthrough of Exercise 23B.1, which introduces the foundational techniques for differentiating polynomial functions. In IB Mathematics, mastering the basic rules of differentiation is essential before moving on to applications such as tangents, kinematic analysis, and optimisation. This article unpacks each rule methodically, provides worked examples, highlights common pitfalls, and extends the concepts to rational exponents and simple motion problems. Whether you are preparing for a test or consolidating your understanding, the bilingual explanations will help you build confidence and accuracy.
欢迎来到练习题 23B.1 的详细讲解。本练习介绍了多项式函数微分的基本技巧。在 IB 数学中，掌握基本的微分法则是后续解决切线、运动分析和优化等应用问题的关键。本文将系统拆解每一条规则，提供详尽的例题，指出常见错误，并将概念扩展到有理指数和简单运动问题。无论是在备考还是巩固理解，双语解释都将帮助你建立信心并提升解题准确性。
1. Understanding Derivatives | 理解导数
The derivative of a function at a point gives the slope of the tangent line to the graph at that point. In physical terms, it represents an instantaneous rate of change – for example, the velocity of a moving object is the derivative of its displacement with respect to time. The process of finding a derivative is called differentiation, and a function is differentiable at a point if the limit of the difference quotient exists.
函数在某一点的导数表示该点切线的斜率。从物理意义上说，它代表瞬时变化率——例如，运动物体的速度就是位移关于时间的导数。求导的过程称为微分，如果在某点差商的极限存在，那么函数在该点可导。
Although the formal limit definition is fundamental, Exercise 23B.1 concentrates on building fluency with differentiation rules that allow us to bypass limits for polynomial functions. Once you can apply these rules mechanically, you will be able to handle more complex expressions with ease.
尽管正式的极限定义是基础，但练习题 23B.1 着重训练多项式函数的微分运算法则，使我们无需每次都计算极限。一旦能够熟练运用这些规则，你就能轻松处理更复杂的表达式。
2. The Power Rule | 幂规则
The most important tool for differentiating polynomials is the power rule. If f(x) = xn, where n is any real number, then f'(x) = n xn–1. In words, you multiply by the exponent, then subtract one from the exponent.
多项式求导中最重要的工具是幂规则。若 f(x) = xⁿ，其中 n 为任意实数，则 f'(x) = n xⁿ⁻¹。含义是：先乘以指数，再将指数减一。
f'(x) = n xⁿ⁻¹
For example, if f(x) = x5, then f'(x) = 5x4. Similarly, the derivative of x3 is 3x2. The power rule also holds for constant functions: the derivative of x0 (= 1) is 0, confirming that the slope of a horizontal line is zero.
例如，若 f(x) = x⁵，则 f'(x) = 5x⁴。同理，x³ 的导数是 3x²。幂规则对常数函数也成立：x⁰（即 1）的导数为 0，验证了水平线的斜率为零。
3. Constant Multiple Rule | 常数倍数法则
When a differentiable function is multiplied by a constant, its derivative is the constant multiplied by the derivative of the function. Stated formally, if y = c · u(x), then dy/dx = c · du/dx.
当一个可导函数乘以常数时，其导数等于该常数乘以函数的导数。形式化表述为：若 y = c · u(x)，则 dy/dx = c · du/dx。
dy/dx = c · u'(x)
For instance, the derivative of y = 7x2 is 7 · 2x1 = 14x. This rule allows us to ‘factor out’ constants before applying the power rule, which simplifies calculations significantly.
例如，y = 7x² 的导数是 7 · 2x¹ = 14x。该法则允许我们先“提取”常数，再应用幂规则，从而大大简化计算。
Be careful: constants that are added or subtracted do not behave in the same way – that requires the sum and difference rules.
注意：被加减的常数不遵循此规则，需要用到和差法则。
4. Sum and Difference Rules | 和差法则
The derivative of a sum of functions is the sum of their derivatives, and the same holds for differences. If f(x) = u(x) ± v(x), then f'(x) = u'(x) ± v'(x). This means we can differentiate polynomials term by term.
函数之和的导数等于各函数导数之和，差的情形同样成立。若 f(x) = u(x) ± v(x)，则 f'(x) = u'(x) ± v'(x)。这意味着我们可以对多项式逐项求导。
Consider f(x) = 4x3 + 2x2 – 5x + 1. Differentiating term by term: the derivative of 4x3 is 12x2, of 2x2 is 4x, of –5x is –5, and of the constant 1 is 0. Hence, f'(x) = 12x2 + 4x – 5.
考虑 f(x) = 4x³ + 2x² – 5x + 1。逐项求导：4x³ 的导数为 12x²，2x² 的导数为 4x，–5x 的导数为 –5，常数 1 的导数为 0。因此，f'(x) = 12x² + 4x – 5。
5. Differentiating Polynomials – Worked Examples | 多项式求导的示例
Combining the power rule, constant multiple rule, and sum/difference rules enables you to differentiate any polynomial smoothly. Let’s work through structured examples to develop fluency.
结合幂规则、常数倍数法则以及和差法则，你就能顺利地对任何多项式求导。我们通过结构化的例题来培养熟练度。
Example 1: Find the derivative of f(x) = 2x4 – 7x3 + 3x2 – x + 6.
例1：求 f(x) = 2x⁴ – 7x³ + 3x² – x + 6 的导数。
Apply the constant multiple and power rules to each term:
Derivative of 2x4 = 2·4x3 = 8x3; of –7x3 = –21x2; of 3x2 = 6x; of –x = –1; of 6 = 0. So f'(x) = 8x3 – 21x2 + 6x – 1.
对每一项应用常数倍数法则和幂规则：2x⁴ 的导数 = 2·4x³ = 8x³；–7x³ 的导数 = –21x²；3x² 的导数 = 6x；–x 的导数 = –1；6 的导数 = 0。因此 f'(x) = 8x³ – 21x² + 6x – 1。
Example 2: Differentiate y = (3x2 + 2)(x – 1). Although this is a product of two functions, you can still use the polynomial rules by expanding first. Expanding gives 3x3 – 3x2 + 2x – 2. Now differentiate term by term: dy/dx = 9x2 – 6x + 2.
例2：求 y = (3x² + 2)(x – 1) 的导数。虽然这是两个函数的乘积，但你可以先展开为多项式再使用规则。展开得 3x³ – 3x² + 2x – 2。然后逐项求导：dy/dx = 9x² – 6x + 2。
6. Notation for Derivatives | 导数的记号
IB Mathematics uses several notations interchangeably for derivatives. The most common are Lagrange notation f'(x), Leibniz notation dy/dx, and the operator form d/dx. Knowing all of them is important because different problems and contexts prefer different notations.
IB 数学中交替使用多种导数记号。最常见的是拉格朗日记号 f'(x)、莱布尼茨记号 dy/dx 以及算子形式 d/dx。熟悉所有记号很重要，因为不同的问题和语境会采用不同的记法。
For example, if y = x4, you can state y’ = 4x3, dy/dx = 4x3, or d/dx (x4) = 4x3. Second derivatives are denoted f”(x) or d2y/dx2.
例如，若 y = x⁴，可表示为 y’ = 4x³、dy/dx = 4x³ 或 d/dx (x⁴) = 4x³。二阶导数记为 f”(x) 或 d²y/dx²。
7. Finding Equations of Tangents | 求切线方程
A direct application of differentiation is determining the equation of the tangent line to a curve at a given point. The derivative gives the gradient m at the specific x-coordinate, and the point-slope form y – y0 = m(x – x0) then yields the tangent equation.
微分的一个直接应用是求曲线上某一点处的切线方程。导数给出了在特定 x 坐标处的斜率 m，再利用点斜式 y – y₀ = m(x – x₀) 即可得到切线方程。
Let f(x) = x2 – 4x + 5 and find the tangent at x = 3. First, f(3) = 9 – 12 + 5 = 2, so the point is (3,2). Then f'(x) = 2x – 4, giving m = 2(3) – 4 = 2. Tangent equation: y – 2 = 2(x – 3), which simplifies to y = 2x – 4.
设 f(x) = x² – 4x + 5，求在 x = 3 处的切线。首先，f(3) = 9 – 12 + 5 = 2，故点为 (3,2)。然后 f'(x) = 2x – 4，得 m = 2(3) – 4 = 2。切线方程为 y – 2 = 2(x – 3)，化简为 y = 2x – 4。
8. Second Derivatives | 二阶导数
The second derivative measures the rate of change of the first derivative. It provides information about the concavity of a graph – whether it is bending upward or downward. In motion problems, the second derivative of displacement gives acceleration.
二阶导数衡量一阶导数的变化率，它提供了函数图像的凹凸性信息——图像是向上弯还是向下弯。在运动问题中，位移的二阶导数给出加速度。
To obtain the second derivative, simply differentiate the first derivative again. For g(x) = 5x3 – 2x2 + x – 4, we have g'(x) = 15x2 – 4x + 1, and then g”(x) = 30x – 4.
要得到二阶导数，只需对一阶导数再次求导即可。对于 g(x) = 5x³ – 2x² + x – 4，有 g'(x) = 15x² – 4x + 1，进而 g”(x) = 30x – 4。
9. Applying Differentiation to Simple Motion | 微分在简单运动中的应用
In kinematics, if displacement s(t) is expressed as a polynomial in time t, then velocity v(t) = s'(t) and acceleration a(t) = v'(t) = s”(t). This transforms a position function into a complete description of motion along a straight line.
在运动学中，若位移 s(t) 表示为时间 t 的多项式，则速度 v(t) = s'(t)，加速度 a(t) = v'(t) = s”(t)。这将位置函数转化为对
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