Example 7.2.2: Differentiation of Polynomials | 例题7.2.2：多项式微分

📚 Example 7.2.2: Differentiation of Polynomials | 例题7.2.2：多项式微分

In this article, we will analyse Example 7.2.2 from the AQA A-Level Mathematics syllabus. This particular example focuses on the differentiation of polynomial functions, a cornerstone of calculus. The problem requires us to apply the power rule accurately and then evaluate the derivative at a given point. We will walk through every stage of the solution in a clear and structured way, including the interpretation of the result and common pitfalls to avoid.

在本文中，我们将分析AQA A-Level数学教学大纲中的例题7.2.2。该例题重点在于多项式函数的微分，这是微积分的基础。题目要求我们准确应用幂法则，然后在给定点处计算导数值。我们将以清晰、结构化的方式逐步展示求解的每一个阶段，包括对结果的解释以及需要避免的常见错误。

1. Understanding Differentiation | 理解微分

Differentiation is a mathematical operation that measures how a function changes as its input changes. For a curve y = f(x), the derivative dy/dx represents the gradient of the tangent at any point on the curve. This gradient tells us the rate at which y increases or decreases with respect to x.

微分是一种数学运算，用于衡量函数随输入变化而变化的速率。对于曲线 y = f(x)，导数 dy/dx 表示曲线上任意一点处切线的斜率。这个斜率告诉我们 y 相对于 x 的增加或减少的速率。

The derivative of a polynomial is obtained by differentiating each term individually. Because polynomials are sums of simple power functions, the process is straightforward once the power rule is mastered. In addition, derivatives are used in many real-world contexts, such as calculating velocity from displacement, finding marginal cost in economics, and optimising shapes in engineering.

多项式的导数可以通过对每一项分别求导来获得。由于多项式是简单幂函数的和，一旦掌握了幂法则，整个过程就变得非常直接；此外，导数在许多现实情境中也有应用，例如由位移计算速度、在经济中求边际成本、在工程中优化形状等。

2. The Power Rule | 幂法则

The power rule states that if f(x) = axⁿ, where a and n are real constants, then the derivative is given by f'(x) = n·a·xⁿ⁻¹. In other words, we bring the exponent down as a multiplier and then reduce the exponent by one.

幂法则指出：若 f(x) = axⁿ，其中 a 和 n 是实常数，则其导数为 f'(x) = n·a·xⁿ⁻¹。换句话说，我们将指数变成乘数，然后使指数减一。

For example, the derivative of 4x³ is 12x². Similarly, the derivative of x⁵ is 5x⁴. When n = 1, the term ax becomes a after differentiation, because x⁰ = 1. When n = 0, the term is a constant, and its derivative is zero. These simple observations allow us to differentiate any polynomial term by term.

例如，4x³ 的导数为 12x²。同样地，x⁵ 的导数为 5x⁴。当 n = 1 时，项 ax 求导后变成 a，因为 x⁰ = 1。当 n = 0 时，这一项就是常数，其导数为零。这些简单结论使我们能够对任何多项式逐项求导。

3. The Example Problem | 例题描述

Example 7.2.2 in the AQA specification asks us to consider the following polynomial:

AQA规范中的例题7.2.2要求我们考虑如下多项式：

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