Mastering Differentiation: First Principles and Core Rules | 掌握微分：第一性原理与核心法则

📚 Mastering Differentiation: First Principles and Core Rules | 掌握微分：第一性原理与核心法则

Differentiation is one of the central pillars of A-Level Mathematics, especially in the Edexcel specification. It allows us to find the rate at which a quantity changes, to locate turning points on curves, and to model motion, economics, and many other applied problems. A strong command of differentiation from first principles and the standard rules is essential for success in both Pure Mathematics and Mechanics.

微分是 A-Level 数学尤其是 Edexcel 考试大纲中的核心支柱之一。它使我们能够求出一个量变化的速率、确定曲线上的转折点，并建立运动学、经济学以及许多其他应用问题的模型。扎实掌握第一性原理求导和标准求导法则，对于纯数学和力学的成功都至关重要。

1. What Is Differentiation? | 什么是微分？

Differentiation is the process of finding the derivative of a function. Geometrically, the derivative at a point gives the gradient of the tangent to the curve at that point. If y = f(x), the derivative is written as f'(x) or dy/dx.

微分是求函数导数的过程。从几何上看，某一点处的导数给出了曲线在该点处切线的斜率。如果 y = f(x)，则导数写作 f'(x) 或 dy/dx。

For a straight line, the gradient is constant. For a curve, the gradient changes from point to point, so we need a function that describes the gradient at every point. The derivative is precisely that function.

对于直线，斜率是恒定的。对于曲线，斜率随点的位置而变化，因此我们需要一个函数来描述每一点处的斜率。导数正是这样的函数。

2. Differentiation from First Principles | 第一性原理求导

The formal definition of the derivative uses a limit. For a function f(x), the derivative f'(x) is defined as the limit of the average rate of change as the interval h tends to zero:

导数的正式定义使用极限。对于函数 f(x)，导数 f'(x) 定义为当区间 h 趋近于零时平均变化率的极限：

f'(x) = lim(h→0) [f(x+h) − f(x)] / h

This expression comes from finding the gradient of a chord between two points on the curve and then letting the two points get infinitely close together.

该表达式来源于先求曲线上两点之间弦的斜率，然后让这两点无限接近。

For example, to differentiate f(x) = x² from first principles, we write:

例如，从第一性原理求 f(x) = x² 的导数，我们可以写出：

f'(x) = lim(h→0) [(x+h)² − x²] / h = lim(h→0) (2xh + h²) / h = 2x

This confirms the standard result that the derivative of x² is 2x. Edexcel exam questions often ask you to carry out this process for simple functions such as x², x³, or 1/x.

这验证了标准结果：x² 的导数是 2x。Edexcel 考试题经常要求你对简单函数如 x²、x³ 或 1/x 执行这一过程。

3. The Power Rule and Polynomials | 幂法则与多项式

In practice, we rarely use first principles for every derivative. The power rule is a shortcut that works for any power of x. If f(x) = xⁿ, then:

实际上，我们很少对每个导数都使用第一性原理。幂法则是一个适用于任意 x 幂次的捷径。如果 f(x) = xⁿ，则：

d/dx (xⁿ) = n xⁿ⁻¹

This means you multiply by the power and then reduce the power by one. For example, the derivative of x⁵ is 5x⁴, and the derivative of x⁻³ is −3x⁻⁴.

这意味着你先乘以幂指数，然后将幂指数减一。例如，x⁵ 的导数是 5x⁴，x⁻³ 的导数是 −3x⁻⁴。

Fractional powers follow the same rule. Since √x can be written as x^½, its derivative is (1/2)x^(−1/2). This is often written as 1/(2√x).

分数幂也遵循同样的法则。由于 √x 可以写成 x^½，它的导数为 (1/2)x^(−1/2)。这通常写作 1/(2√x)。

Function y \| 函数 y	Derivative dy/dx \| 导数 dy/dx
x³	3x²
x⁷	7x⁶
x⁻²	−2x⁻³
√x = x^½	(1/2)x^(−1/2)

4. Constant Multiple and Sum/Difference Rules | 常数倍与和差法则

Differentiation is linear, which means constants can be taken outside the derivative and sums can be differentiated term by term. If k is a constant, then:

微分是线性的，这意味着常数可以移到导数符号外，和式可以逐项求导。如果 k 是常数，则：

d/dx [k f(x)] = k f'(x)

Similarly, for two functions u(x) and v(x):

类似地，对于两个函数 u(x) 和 v(x)：

d/dx [u(x) ± v(x)] = u'(x) ± v'(x)

These rules allow us to differentiate any polynomial quickly. For example, if y = 4x³ − 2x² + 5x − 7, then dy/dx = 12x² − 4x + 5. Notice that constant terms always differentiate to zero.

这些法则使我们能够快速地对任何多项式求导。例如，如果 y = 4x³ − 2x² + 5x − 7，则 dy/dx = 12x² − 4x + 5。注意常数项的导数总是零。

5. The Product Rule | 乘法法则

When differentiating a product of two functions, we cannot simply multiply the derivatives. Instead, if y = u(x)v(x), then:

当对两个函数的乘积求导时，不能简单地将导数相乘。相反，如果 y = u(x)v(x)，则：

dy/dx = u (dv/dx) + v (du/dx)

This is called the product rule. It is usually remembered as “first times derivative of second plus second times derivative of first”.

这称为乘法法则。通常记忆为“第一个函数乘第二个函数的导数，加上第二个函数乘第一个函数的导数”。

For example, if y = x² sin x, let u = x² and v = sin x. Then du/dx = 2x and dv/dx = cos x. Using the product rule:

例如，如果 y = x² sin x，令 u = x²，v = sin x。则 du/dx = 2x，dv/dx = cos x。使用乘法法则：

dy/dx = x² cos x + 2x sin x

Edexcel questions often require the product rule with trigonometric, exponential, or logarithmic functions, and sometimes you must combine it with the chain rule.

Edexcel 考题通常要求在三角函数、指数函数或对数函数中使用乘法法则，有时还必须将其与链式法则结合使用。

6. The Quotient Rule | 除法法则

The quotient rule is used when differentiating a function divided by another function. If y = u(x) / v(x), then:

当一个函数除以另一个函数时，使用除法法则求导。如果 y = u(x) / v(x)，则：

dy/dx = [v (du/dx) − u (dv/dx)] / v²

Notice the minus sign in the numerator. Many students lose marks by forgetting the order or missing the negative sign. It is often helpful to label u, v, du/dx, and dv/dx before substituting.

注意分子中的减号。许多学生因为顺序写错或漏掉负号而失分。在代入之前先标出 u、v、du/dx 和 dv/dx 会很有帮助。

For example, if y = x / (x + 1), then u = x, v = x + 1, du/dx = 1, and dv/dx = 1. The quotient rule gives:

例如，如果 y = x / (x + 1)，则 u = x，v = x + 1，du/dx = 1，dv/dx = 1。除法法则给出：

dy/dx = [(x + 1)(1) − x(1)] / (x + 1)² = 1 / (x + 1)²

This result appears frequently in integration topics as well, so being able to differentiate rational functions confidently is very valuable.

这一结果在积分专题中也经常出现，因此能够自信地对有理函数求导非常有价值。

7. The Chain Rule | 链式法则

The chain rule is used for composite functions, that is, functions of functions. If y is a function of u and u is a function of x, then:

链式法则用于复合函数，即函数的函数。如果 y 是 u 的函数，而 u 是 x 的函数，则：

dy/dx = (dy/du) × (du/dx)

This rule is essential for differentiating expressions such as (3x + 2)⁵, sin(2x), or e^(x²). You differentiate the outer function and then multiply by the derivative of the inner function.

该法则对于求 (3x + 2)⁵、sin(2x) 或 e^(x²) 等表达式的导数至关重要。你先对外层函数求导，然后乘以内层函数的导数。

For example, if y = (3x + 2)⁵, let u = 3x + 2. Then y = u⁵, dy/du = 5u⁴, and du/dx = 3. Therefore:

例如，如果 y = (3x + 2)⁵，令 u = 3x + 2。则 y = u⁵，dy/du = 5u⁴，du/dx = 3。因此：

dy/dx = 5u⁴ × 3 = 15(3x + 2)⁴

In Edexcel exams, the chain rule is often tested inside product or quotient rule problems, so it is important to recognise when a function is composite.

在 Edexcel 考试中，链式法则经常被嵌套在乘法法则或除法法则的问题中，因此识别一个函数是否为复合函数非常重要。

8. Higher Derivatives | 高阶导数

Differentiating a function once gives the first derivative. Differentiating the first derivative gives the second derivative, written as f”(x) or d²y/dx². Higher derivatives are used to classify stationary points and to model acceleration in mechanics.

对函数求导一次得到一阶导数。对一阶导数再求导得到二阶导数，写作 f”(x) 或 d²y/dx²。高阶导数用于判断驻点类型以及在力学中建立加速度模型。

For example, if y = x⁴ − 3x² + 2x, then the first derivative is dy/dx = 4x³ − 6x + 2. Differentiating again gives the second derivative d²y/dx² = 12x² − 6.

例如，如果 y = x⁴ − 3x² + 2x，则一阶导数为 dy/dx = 4x³ − 6x + 2。再次求导得到二阶导数 d²y/dx² = 12x² − 6。

In Mechanics, if displacement is denoted by s, then velocity is ds/dt and acceleration is d²s/dt². This connection is heavily examined in Edexcel Mechanics modules.

在力学中，如果位移用 s 表示，那么速度为 ds/dt，加速度为 d²s/dt²。这一联系在 Edexcel 力学模块中被大量考查。

9. Stationary Points and Turning Points | 驻点与转折点

A stationary point occurs where the gradient of a curve is zero, so dy/dx = 0. There are three main types: local maximum, local minimum, and point of inflection. To classify them, we often use the second derivative test.

驻点出现在曲线的斜率为零的位置，即 dy/dx = 0。驻点主要有三种类型：局部最大值、局部最小值和拐点。我们通常使用二阶导数检验来分类。

If d²y/dx² is positive at a stationary point, the curve is concave up and the point is a local minimum. If d²y/dx² is negative, the curve is concave down and the point is a local maximum. If d²y/dx² = 0, further investigation is needed.

如果驻点处的 d²y/dx² 为正，曲线向上凹，该点为局部最小值；如果 d²y/dx² 为负，曲线向下凹，该点为局部最大值；如果 d²y/dx² = 0，则需要进一步检验。

For example, for y = x³ − 3x, the derivative is dy/dx = 3x² − 3. Setting this equal to zero gives x = ±1. The second derivative is d²y/dx² = 6x. At x = 1, the second derivative is positive, so there is a minimum. At x = −1, the second derivative is negative, so there is a maximum.

例如，对于 y = x³ − 3x，导数为 dy/dx = 3x² − 3。令其等于零可得 x = ±1。二阶导数为 d²y/dx² = 6x。在 x = 1 处二阶导数为正，因此存在极小值；在 x = −1 处二阶导数为负，因此存在极大值。

10. Real Exam Tips for Edexcel Differentiation | Edexcel 微分实考技巧

In Edexcel A-Level Maths, differentiation questions often combine several rules in one problem. Here are some key points to remember:

在 Edexcel A-Level 数学中，微分题经常在一个问题中综合多个法则。以下是一些需要记住的要点：

Always simplify the function before differentiating when possible, especially with powers and roots. | 在可能的情况下，求导前先化简函数，尤其是含有幂和根式时。
Label u and v clearly in product and quotient rule problems to avoid sign errors. | 在乘法法则和除法法则问题中，清晰地标出 u 和 v，以避免符号错误。
Check whether a function is composite and apply the chain rule when necessary. | 检查函数是否为复合函数，并在必要时应用链式法则。
Do not forget that the derivative of a constant is zero. | 不要忘记常数的导数为零。
When classifying stationary points, always show either the second derivative test or a sign change test. | 在判断驻点类型时，要始终展示二阶导数检验或符号变化检验的过程。

Mastering differentiation rules gives you a strong foundation for integration, differential equations, and applied modelling, all of which feature heavily in the Edexcel specification.

掌握微分法则为积分、微分方程和应用建模奠定了坚实的基础，这些都是 Edexcel 考试大纲中的重要内容。

Published by TutorHao | Edexcel A-Level Maths Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

Mastering Differentiation: First Principles and Core Rules | 掌握微分：第一性原理与核心法则