📚 Differentiation: First Principles and Key Techniques | 微分：第一性原理与核心技巧
Differentiation is one of the central pillars of A-Level Mathematics. It gives us a precise way to measure how a function changes at any instant, and it forms the basis for optimisation, curve sketching, kinematics and many applied problems. In the Edexcel specification, you are expected to understand differentiation from first principles, apply standard rules fluently, and use the derivative to analyse gradients, tangents, normals, stationary points and the shape of a curve.
微分是 A-Level 数学的核心支柱之一。它为我们提供了一种精确的方法来衡量函数在任意瞬间的变化，并构成了最优化、曲线作图、运动学以及许多应用问题的基础。在 Edexcel 考试大纲中，你需要理解从第一性原理出发的微分，熟练应用标准法则，并利用导数分析梯度、切线、法线、驻点以及曲线的形状。
1. Why Differentiation Matters | 为什么微分重要
Differentiation answers a fundamental question: how steep is a curve at a particular point? For a straight line, the gradient is constant, but for a curve the gradient changes continuously. The derivative gives the gradient of the tangent to the curve at any chosen point, so it is often called the gradient function.
微分回答了一个基本问题：曲线在某一点的陡峭程度如何？对于直线，梯度是恒定的；但对于曲线，梯度在不断变化。导数给出了曲线在任意选定点处切线的梯度，因此它通常被称为梯度函数。
This idea appears throughout A-Level Mathematics: in pure mathematics when finding maximum and minimum values, in mechanics when velocity is the derivative of displacement, and in statistics when approximating rates of change. Mastering differentiation therefore unlocks many other topics.
这一思想贯穿于 A-Level 数学的始终：在纯数学中求最大值和最小值，在力学中速度是位移的导数，在统计中近似变化率。因此，掌握微分可以打开许多其他主题的大门。
2. Gradient of a Curve and Limits | 曲线的梯度与极限
To find the gradient of a curve at a point, we cannot simply use two fixed points as we would for a straight line. Instead, we take two points very close together and let the distance between them shrink towards zero. This process is called taking a limit.
要找到曲线上某一点的梯度，我们不能像求直线那样简单地使用两个固定点。相反，我们取两个非常接近的点，并让它们之间的距离趋近于零。这个过程称为取极限。
If A(x, f(x)) and B(x+h, f(x+h)) are two points on the curve y = f(x), the gradient of the chord AB is:
如果 A(x, f(x)) 和 B(x+h, f(x+h)) 是曲线 y = f(x) 上的两个点，那么弦 AB 的梯度为：
Gradient of chord = [f(x+h) − f(x)] / h
As h becomes smaller, the chord gets closer and closer to the tangent at A. The gradient of the tangent is therefore the limit of this expression as h tends to 0.
随着 h 越来越小，弦越来越接近 A 点处的切线。因此，切线的梯度就是当 h 趋于 0 时该表达式的极限。
3. First Principles Formula | 第一性原理公式
The formal definition of the derivative is the limit of the difference quotient. For a function f, the derivative f'(x) is defined by:
导数的正式定义是差商的极限。对于函数 f，导数 f'(x) 定义为：
f ‘(x) = lim (h → 0) [f(x+h) − f(x)] / h
This expression is called differentiation from first principles. The notation dy/dx is equivalent and is read as ‘the derivative of y with respect to x’. Both notations are used in Edexcel exams, so you should be comfortable switching between them.
这个表达式称为从第一性原理出发的微分。记号 dy/dx 与之等价，读作“y 对 x 的导数”。Edexcel 考试中两种记号都会使用，因此你应该能够熟练地在它们之间切换。
When using first principles, always write out the full expansion of f(x+h) before simplifying. A common mistake is to cancel h too early or to forget that the limit is taken only after simplification.
使用第一性原理时，一定要先完整写出 f(x+h) 的展开式，然后再化简。常见的错误是过早约去 h，或者忘记极限只能在化简之后才取。
4. Worked First Principles Example: f(x) = x² | 第一性原理示例：f(x) = x²
Let f(x) = x². We need to find f'(x) from the definition. Start by computing f(x+h):
设 f(x) = x²。我们需要根据定义求 f'(x)。先计算 f(x+h)：
f(x+h) = (x+h)² = x² + 2xh + h²
Now form the difference quotient:
现在构造差商：
[f(x+h) − f(x)] / h = [x² + 2xh + h² − x²] / h = [2xh + h²] / h
Since h is not zero, we can simplify by cancelling h:
由于 h 不等于零，我们可以约去 h 来化简：
[2xh + h²] / h = 2x + h
Finally, take the limit as h tends to 0:
最后，对 h 趋于 0 取极限：
f ‘(x) = lim (h → 0) (2x + h) = 2x
This confirms the familiar result that the derivative of x² is 2x. In an exam, you must show all of these steps clearly to earn full marks for a first principles question.
这验证了我们熟悉的结果：x² 的导数是 2x。在考试中，你必须清晰地展示所有这些步骤，才能在第一性原理题目中获得满分。
5. Power Rule and Constant Multiple Rule | 幂法则与常数倍法则
Once the first principles idea is understood, you can use faster rules for differentiation. The power rule states that for any real constant n:
一旦理解了第一性原理的思想，你就可以使用更快的微分法则。幂法则指出，对于任意实数常数 n：
If f(x) = xⁿ, then f ‘(x) = nxⁿ⁻¹
For example, if y = x⁵ then dy/dx = 5x⁴. If y = x⁻² then dy/dx = −2x⁻³. If y = x^(1/2) then dy/dx = (1/2)x^(−1/2).
例如，如果 y = x⁵，那么 dy/dx = 5x⁴。如果 y = x⁻²，那么 dy/dx = −2x⁻³。如果 y = x^(1/2)，那么 dy/dx = (1/2)x^(−1/2)。
The constant multiple rule says that a constant multiplying a function can be kept outside the differentiation. If y = k f(x) where k is a constant, then dy/dx = k f'(x).
常数倍法则指出，乘以函数的常数可以保留在微分之外。如果 y = k f(x)，其中 k 是常数，那么 dy/dx = k f'(x)。
These two rules together allow you to differentiate expressions such as 7x³ quickly: the derivative is 7 × 3x² = 21x².
这两条法则结合起来，可以让你快速对诸如 7x³ 这样的表达式求导：导数为 7 × 3x² = 21x²。
6. Sum and Difference Rules | 和差法则
Differentiation is a linear operation, which means we can differentiate term by term. If f(x) and g(x) are differentiable functions, then:
微分是一种线性运算，这意味着我们可以逐项求导。如果 f(x) 和 g(x) 都是可微函数，那么：
(f ± g)’ = f ‘ ± g ‘
In Leibniz notation, if y = u + v − w, then dy/dx = du/dx + dv/dx − dw/dx. This is extremely useful for polynomials, which are simply sums of power functions.
在莱布尼茨记号中，如果 y = u + v − w，那么 dy/dx = du/dx + dv/dx − dw/dx。这对多项式非常有用，因为多项式就是幂函数的和。
For example, differentiate y = 3x⁴ − 5x² + 2x − 7. Applying the power rule and sum rule term by term gives:
例如，求 y = 3x⁴ − 5x² + 2x − 7 的导数。逐项应用幂法则与和差法则得到：
dy/dx = 12x³ − 10x + 2
Note that the derivative of a constant term is zero. This is because a constant function has zero rate of change.
注意，常数项的导数为零。这是因为常数函数的变化率为零。
7. Differentiating Polynomials | 多项式求导
A polynomial is a sum of terms of the form axⁿ. To differentiate a polynomial, apply the power rule to each term and keep the constants where they are. This is usually the first skill assessed in an Edexcel differentiation question.
多项式是由 axⁿ 形式的项组成的和。对多项式求导时，对每一项应用幂法则，并保留常数。这通常是 Edexcel 微分题中考查的第一项技能。
Consider y = 2x⁵ − 3x³ + 4x² − x + 9. Differentiating term by term:
考虑 y = 2x⁵ − 3x³ + 4x² − x + 9。逐项求导：
dy/dx = 10x⁴ − 9x² + 8x − 1
Always rewrite terms with negative or fractional powers before differentiating. For example, y = 1/x² becomes y = x⁻², and y = √x becomes y = x^(1/2). Then the power rule applies directly.
在求导前，一定要先把负指数或分数指数项改写过来。例如，y = 1/x² 改写为 y = x⁻²，y = √x 改写为 y = x^(1/2)。然后就能直接应用幂法则。
This technique is especially important when differentiating rational functions or expressions involving roots, which appear frequently on Edexcel papers.
这种技巧在求有理函数或含根式的表达式的导数时尤其重要，这类题目在 Edexcel 试卷中经常出现。
8. Tangents and Normals | 切线与法线
Once you have the derivative, you can find the equation of the tangent or normal to a curve at a given point. The tangent is the line that just touches the curve, and its gradient is the value of dy/dx at that point.
一旦求出了导数，你就可以找到曲线在给定点处的切线或法线方程。切线是与曲线只有一个接触点的直线，其梯度就是该点处 dy/dx 的值。
To find a tangent, first substitute the x-coordinate into dy/dx to get the gradient m. Then use the point-slope form of a straight line:
要求切线，首先将 x 坐标代入 dy/dx 得到梯度 m。然后使用直线的点斜式：
y − y₁ = m(x − x₁)
The normal is perpendicular to the tangent, so its gradient is the negative reciprocal of m. If the tangent gradient is m, the normal gradient is −1/m, provided m ≠ 0.
法线垂直于切线，因此其梯度是 m 的负倒数。如果切线梯度为 m，则法线梯度为 −1/m，前提是 m ≠ 0。
For example, find the tangent to y = x³ at x = 2. First, dy/dx = 3x², so at x = 2 the gradient is 12. The point is (2, 8), so the tangent is:
例如，求 y = x³ 在 x = 2 处的切线。首先 dy/dx = 3x²，因此在 x = 2 处梯度为 12。点为 (2, 8)，所以切线为：
y − 8 = 12(x − 2), or y = 12x − 16
In exam questions, you may need to leave your answer in the form ax + by + c = 0, so practise rearranging confidently.
在考试题中，你可能需要将答案写成 ax + by + c = 0 的形式，因此要练习熟练地进行整理。
9. Second Derivative and Concavity | 二阶导数与凹凸性
The second derivative is simply the derivative of the derivative. If y = f(x), then the first derivative is dy/dx = f'(x), and the second derivative is d²y/dx² = f”(x). It measures the rate of change of the gradient.
二阶导数就是导数的导数。如果 y = f(x)，那么一阶导数为 dy/dx = f'(x)，二阶导数为 d²y/dx² = f”(x)。它衡量的是梯度的变化率。
The sign of the second derivative tells you about the curvature of a graph. If f”(x) > 0 on an interval, the curve is concave up, meaning it bends upwards like a smile. If f”(x) < 0, the curve is concave down, like a frown.
二阶导数的符号告诉你图形的弯曲情况。如果在某个区间上 f”(x) > 0，曲线是凹向上的，也就是说它像微笑一样向上弯曲。如果 f”(x) < 0，曲线是凹向下的，像皱眉一样。
This information is useful for determining the nature of stationary points and for understanding the overall shape of a curve before sketching it.
这些信息对于判断驻点的性质以及在作图前理解曲线的整体形状都非常有用。
10. Increasing and Decreasing Functions | 递增与递减函数
A function is increasing on an interval if its derivative is positive throughout that interval. Similarly, it is decreasing if its derivative is negative. This is a direct consequence of the derivative measuring gradient.
如果一个函数在某个区间上的导数始终为正，那么它在该区间上是递增的。同样，如果导数为负，那么它是递减的。这是导数衡量梯度的直接结果。
If f'(x) > 0 for all x in (a, b), then f is increasing on (a, b). If f'(x) < 0 for all x in (a, b), then f is decreasing on (a, b).
如果对于 (a, b) 内的所有 x 都有 f'(x) > 0，那么 f 在 (a, b) 上递增。如果对于 (a, b) 内的所有 x 都有 f'(x) < 0，那么 f 在 (a, b) 上递减。
To determine where a function is increasing or decreasing, find f'(x), solve f'(x) = 0 to find critical points, and then test the sign of f'(x) in the intervals around those points.
要确定函数在哪里递增或递减，先求 f'(x)，解 f'(x) = 0 找到临界点，然后在临界点两侧的区间中检验 f'(x) 的符号。
This sign chart method is an essential skill for Edexcel questions on curve behaviour and stationary point classification.
这种符号表方法是 Edexcel 曲线行为与驻点分类题目中的一项基本技能。
11. Stationary Points and Classification | 驻点及其分类
Stationary points occur where the gradient is zero, that is where f'(x) = 0. There are three main types: local maximum, local minimum, and stationary point of inflection. Classifying them is a common exam requirement.
驻点出现在梯度为零的位置，即 f'(x) = 0 的地方。驻点主要有三种类型：局部最大值、局部最小值和驻点拐点。对驻点进行分类是常见的考试要求。
At a local maximum, the gradient changes from positive to negative as x increases. At a local minimum, the gradient changes from negative to positive. At a stationary point of inflection, the gradient does not change sign.
在局部最大值处，随着 x 增大，梯度从正变为负。在局部最小值处，梯度从负变为正。在驻点拐点处，梯度的符号不变。
There are two main methods for classification. The first uses the sign of f'(x) on either side of the stationary point. The second uses the second derivative: if f”(x) > 0 it is a minimum, if f”(x) < 0 it is a maximum, and if f''(x) = 0 the test is inconclusive.
分类有两种主要方法。第一种使用驻点两侧 f'(x) 的符号。第二种使用二阶导数：如果 f”(x) > 0 则为最小值，如果 f”(x) < 0 则为最大值，如果 f''(x) = 0 则该方法无法判断。
For example, y = x³ − 3x has f'(x) = 3x² − 3 = 3(x² − 1). Setting f'(x) = 0 gives x = 1 and x = −1. Since f”(x) = 6x, at x = 1, f”(1) = 6 > 0, so (1, −2) is a minimum; at x = −1, f”(−1) = −6 < 0, so (−1, 2) is a maximum.
例如，y = x³ − 3x 的 f'(x) = 3x² − 3 = 3(x² − 1)。令 f'(x) = 0 得 x = 1 和 x = −1。因为 f”(x) = 6x，在 x = 1 处 f”(1) = 6 > 0，所以 (1, −2) 是最小值；在 x = −1 处 f”(−1) = −6 < 0，所以 (−1, 2) 是最大值。
12. Common Mistakes and Exam Tips | 常见错误与考试技巧
One frequent mistake in first principles questions is failing to simplify the numerator before taking the limit. Always expand and cancel h completely, then substitute h = 0.
第一性原理题中一个常见错误是在取极限前没有化简分子。一定要先完整展开并约去 h，然后再代入 h = 0。
Another common error is forgetting that the derivative of a constant is zero. In longer expressions, students sometimes try to differentiate a constant as 1, which is incorrect.
另一个常见错误是忘记常数的导数为零。在较长的表达式中，有些学生有时会把常数误求导为 1，这是不正确的。
When finding tangents and normals, always check whether the question wants the tangent or the normal. A normal gradient is the negative reciprocal of the tangent gradient, not the same.
在求切线和法线时，一定要看清题目要求的是切线还是法线。法线梯度是切线梯度的负倒数，而不是相同。
In stationary point problems, show a clear sign table or second derivative conclusion. Simply stating ‘maximum’ without justification will not earn full marks.
在驻点问题中，要展示清晰的符号表或二阶导数结论。仅仅写出“最大值”而没有理由是不能得到满分的。
Finally, when differentiating terms with roots or fractions, always rewrite them as powers first. This reduces errors and makes the power rule straightforward.
最后，在求含根式或分式的项的导数时，一定要先把它们改写为幂的形式。这样可以减少错误，并使幂法则直接适用。
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