📚 Differentiation from First Principles and Basic Rules | 从第一性原理出发的微分法及基本求导法则
Differentiation is one of the foundation skills in Edexcel A Level Mathematics. It measures how a function changes as its input changes, and it is needed for tangents, normals, optimisation and curve sketching.
微分是 Edexcel A Level 数学的基础技能之一。它度量函数值随自变量变化而变化的速率，并且是求切线、法线、优化问题和曲线草图所必需的。
1. What is Differentiation? | 什么是微分
Differentiation is the process of finding the gradient function of a curve. For a straight-line graph y = mx + c, the gradient is constant, but for a curve the gradient changes at every point.
微分是求曲线斜率函数的过程。对于直线 y = mx + c，斜率是常数，但对于曲线，每一点的斜率都在变化。
Differentiation gives a new function f ′(x) that tells us the instantaneous rate of change of f(x) with respect to x. This is why the derivative is often called the gradient function.
微分给出了一个新函数 f ′(x)，它告诉我们 f(x) 关于 x 的瞬时变化率。这就是为什么导数通常被称为斜率函数。
2. The First-Principles Formula | 第一性原理公式
The derivative can be defined from first principles using the gradient of a chord between two points close together. If the point is (x, f(x)) and a nearby point is (x + h, f(x + h)), then the chord gradient is given by the difference quotient.
导数可以用第一性原理来定义，即两个非常接近的点之间的弦的斜率。取点 (x, f(x)) 和邻近点 (x + h, f(x + h))，弦的斜率由差商给出。
As h tends to zero, the chord becomes the tangent, so the limit is the derivative. This definition is central to Edexcel AS Pure Mathematics and is often tested directly.
当 h 趋向于零时，弦变成切线，因此这个极限就是导数。这个定义是 Edexcel AS 纯数学的核心内容，并且经常被直接考查。
f ′(x) = lim (h → 0) [ f(x + h) − f(x) ] / h
3. Worked Example: f(x) = x² | 例题：f(x) = x²
To differentiate f(x) = x² from first principles, we substitute f(x + h) = (x + h)² into the formula.
要从第一性原理出发对 f(x) = x² 求导，我们将 f(x + h) = (x + h)² 代入公式。
Simplify (x + h)² − x² as x² + 2xh + h² − x² = 2xh + h². Dividing by h gives 2x + h, and as h → 0 the limit is 2x.
将 (x + h)² − x² 化简为 x² + 2xh + h² − x² = 2xh + h²。除以 h 得到 2x + h，当 h → 0 时极限为 2x。
f ′(x) = lim (h → 0) [ (x + h)² − x² ] / h = 2x
This shows that the gradient of y = x² at any point x is exactly 2x. It is a good idea to write every algebraic step clearly in Edexcel exam questions.
这表明 y = x² 在任意点 x 处的斜率恰好是 2x。在 Edexcel 考试题中，最好清晰地写出每一步代数运算。
4. Generalising to f(x) = xⁿ | 推广到 f(x) = xⁿ
Repeating the same process for f(x) = x³ gives f ′(x) = 3x². For f(x) = x⁴, the derivative is 4x³. This pattern leads to the power rule for differentiation.
对 f(x) = x³ 重复同样的过程会得到 f ′(x) = 3x²。对 f(x) = x⁴，其导数为 4x³。这一规律引出了微分的幂法则。
In general, for any real constant n, the power rule states that the derivative of xⁿ is nxⁿ⁻¹. This rule works for positive, negative and fractional powers, which is very useful across the Edexcel specification.
一般来说，对于任意实常数 n，幂法则指出 xⁿ 的导数是 nxⁿ⁻¹。该法则适用于正指数、负指数和分数指数，在整个 Edexcel 考试大纲中都非常有用。
d/dx (xⁿ) = nxⁿ⁻¹
5. Notation for Derivatives | 导数记号
Edexcel questions may use several notations for the derivative. The most common are f ′(x), y′, dy/dx and d/dx [ f(x) ]. All of these mean the same thing.
Edexcel 题目可能会使用几种导数记号。最常见的是 f ′(x)、y′、dy/dx 和 d/dx [ f(x) ]。它们都表示相同的含义。
When the derivative is evaluated at a specific point x = a, we write f ′(a) or dy/dx at x = a. Make sure you can move confidently between these notations.
当导数在特定点 x = a 处求值时，我们写作 f ′(a) 或 dy/dx 在 x = a 处。确保你能在这些记号之间自如转换。
6. Basic Differentiation Rules | 基本求导法则
Differentiation is linear, so constants, multiples and sums can be handled term by term. The constant rule says that the derivative of a constant is zero.
微分是线性的，因此常数、倍数和和可以逐项处理。常数法则指出，常数的导数为零。
For example, d/dx [5x³ − 2x + 7] = 15x² − 2, because the derivative of the constant 7 is 0. You do not need to use first principles every time once these rules are known.
例如，d/dx [5x³ − 2x + 7] = 15x² − 2，因为常数 7 的导数是 0。一旦掌握了这些法则，就不需要每次都使用第一性原理。
d/dx [ a f(x) + b g(x) ] = a f ′(x) + b g ′(x)
7. Tangents and Normals | 切线与法线
The tangent to y = f(x) at x = a has gradient f ′(a). Its equation is y − f(a) = f ′(a)(x − a). This is a standard exam application of differentiation.
y = f(x) 在 x = a 处的切线斜率为 f ′(a)。其方程为 y − f(a) = f ′(a)(x − a)。这是微分在考试中的标准应用。
The normal is perpendicular to the tangent, so its gradient is the negative reciprocal of the tangent gradient. If the tangent gradient is f ′(a), the normal gradient is −1 / f ′(a), provided f ′(a) is not zero.
法线垂直于切线，因此它的斜率是切线斜率的负倒数。如果切线斜率为 f ′(a)，则法线斜率为 −1 / f ′(a)，前提是 f ′(a) 不为零。
Tangent gradient = f ′(a), normal gradient = −1 / f ′(a)
8. Stationary Points | 驻点
A stationary point occurs where the tangent is horizontal, so we solve f ′(x) = 0. This gives the x-coordinates of possible maximum, minimum or inflection points.
驻点出现在切线为水平的位置，因此我们解方程 f ′(x) = 0。这会给出可能的极大值点、极小值点或拐点的 x 坐标。
Stationary points are central to optimisation problems in Edexcel A Level Mathematics. You should always find the y-coordinate by substituting back into the original function f(x).
驻点是 Edexcel A Level 数学中优化问题的核心。你应该始终通过代入原函数 f(x) 来求得 y 坐标。
f ′(x) = 0
9. Second Derivative and Nature | 二阶导数与驻点性质
The second derivative f ′′(x) is the derivative of f ′(x). It measures the rate of change of the gradient and helps determine the nature of a stationary point.
二阶导数 f ′′(x) 是 f ′(x) 的导数。它度量斜率的变化率，并有助于判断驻点的性质。
At a stationary point x = a, if f ′′(a) > 0, the point is a local minimum. If f ′′(a) < 0, the point is a local maximum. If f ′′(a) = 0, you should use a sign-change test on f ′(x) around a.
在驻点 x = a 处，如果 f ′′(a) > 0，该点是局部极小值点；如果 f ′′(a) < 0，该点是局部极大值点。如果 f ′′(a) = 0，你应该在 a 的附近对 f ′(x) 使用符号变化检验。
10. Common Mistakes and Exam Tips | 常见错误与考试提示
Common pitfalls include forgetting to differentiate constants, mishandling negative or fractional powers, and using the normal gradient incorrectly.
常见错误包括忘记对常数求导、错误处理负指数或分数指数，以及错误使用法线斜率。
Always simplify algebra before differentiating where possible. When a question says ‘using first principles’, you must show the full limit process, not just quote the power rule.
在可能的情况下，求导前先化简代数式。当题目说 ‘using first principles’ 时，你必须展示完整的极限过程，而不能只引用幂法则。
Practise switching between forms such as 1/x² = x⁻² and √x = x^½, because Edexcel papers frequently test these conversions before differentiation.
练习在诸如 1/x² = x⁻² 和 √x = x^½ 等形式之间转换，因为 Edexcel 试卷经常在求导前考查这些转换。
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