📚 Differentiation | 微分
Differentiation is one of the two central pillars of calculus, providing a powerful tool to analyse how functions change. In the Edexcel A-Level Mathematics specification, it appears across Pure 1, Pure 2, Pure 3, and Pure 4, progressively building from the gradient of a secant line to advanced techniques like implicit differentiation and connected rates of change.
微分是微积分的两大核心支柱之一，为分析函数的变化提供了强大工具。在 Edexcel A-Level 数学考纲中，微分贯穿 Pure 1、Pure 2、Pure 3 和 Pure 4 各个模块，从割线的斜率逐步深入到隐函数微分和相关变化率等高级技巧。
1. The Gradient of a Curve | 曲线的梯度
The gradient of a straight line is constant, but a curve's steepness changes from point to point. Differentiation allows us to find this instantaneous rate of change by considering the limiting value of the slope of a chord as the two points approach each other.
直线的梯度是恒定的，但曲线上每一点的陡峭程度各不相同。通过考虑当弦的两个端点无限接近时弦斜率的极限值，微分可以帮助我们找到这种瞬时变化率。
g the curve towards P, and the chord gradient approaches the gradient of the tangent at P.
如果在曲线 y = f(x) 上取两点 P(x, f(x)) 和 Q(x+h, f(x+h))，弦 PQ 的斜率为 [f(x+h) − f(x)] / h。当 h → 0 时，Q 沿曲线趋近于 P，弦的斜率就趋近于曲线在 P 点切线的斜率。
2. Differentiation from First Principles | 导数的第一原理
The formal definition of the derivative uses the limit process described above. For a function f(x), its derivative f'(x) is given by:
导数的正式定义采用了上述极限过程。对于函数 f(x)，其导数 f'(x) 表示为：
f '(x) = lim (h → 0) [f(x + h) − f(x)] / h
This is called differentiating from first principles. It is essential for understanding the underlying concept, and Edexcel often expects students to apply it to simple functions such as x².
这被称为从第一原理求导，是理解微分基本概念的关键。Edexcel 考试经常要求学生将此方法应用于 x² 等简单函数。
f(x)] / h = (2xh + h²)/h = 2x + h. As h → 0, this approaches 2x, so the derivative is 2x.
对于 f(x) = x²，展开 f(x+h) = (x+h)² = x² + 2xh + h²。那么 [f(x+h) − f(x)] / h = (2xh + h²)/h = 2x + h。当 h → 0 时，该式趋近于 2x，因此导数为 2x。
3. Basic Rules of Differentiation | 微分的基本法则
From first principles, we obtain a simple power rule for differentiating xⁿ, where n is any real number. This rule states:
由第一原理推导出一个简单的幂函数法则，可用于对 xⁿ 求导，其中 n 为任意实数。法则如下：
If y = xⁿ, then dy/dx = n xⁿ⁻¹
constant function y = c, the derivative is 0 because a horizontal line has zero gradient.
例如，若 y = x⁵，则 dy/dx = 5x⁴。若 y = 1/x = x⁻¹，则 dy/dx = −1 x⁻² = −1/x²。对于常函数 y = c，导数为 0，因为水平线的斜率为零。
ultiplied by a constant or expressed as a sum of terms, differentiation is linear: (kf(x))' = k f '(x) and (f(x) + g(x))' = f '(x) + g'(x). Thus, polynomials can be differentiated term by term.
当函数乘以常数或表示为多项之和时，微分运算具有线性性质：(kf(x))' = k f '(x) 且 (f(x) + g(x))' = f '(x) + g'(x)。因此，多项式可以逐项求导。
4. The Chain Rule | 链式法则
), then the derivative is the product of the derivative of the outer function evaluated at the inner function and the derivative of the inner function:
链式法则用于对复合函数求导。若 y = f(g(x))，则导数等于外层函数在内层函数处的导数乘以内层函数的导数：
dy/dx = f '(g(x)) × g'(x)
). This is especially useful when dealing with powers of linear functions, such as (3x + 2)⁵.
一种便捷形式是令 u = g(x)，则 y = f(u)。于是 dy/dx = (dy/du) × (du/dx)。在处理如 (3x+2)⁵ 这样的线性函数幂时，这种方法特别有用。
5(3x+2)⁴ × 3 = 15(3x+2)⁴. The chain rule can be extended to multiple layers, such as y = esin(x²).
对于 y = (3x+2)⁵，设 u = 3x+2，则 y = u⁵，dy/du = 5u⁴，du/dx = 3，于是 dy/dx = 5(3x+2)⁴ × 3 = 15(3x+2)⁴。链式法则可推广到多层复合，例如 y = esin(x²)。
5. The Product Rule | 乘积法则
ivative is not simply the product of their individual derivatives. If y = u(x) v(x), then:
当两个函数相乘时，导数并不是各自导数的简单乘积。若 y = u(x) v(x)，则有：
dy/dx = u dv/dx + v du/dx
An easy way to remember is 'first times derivative of second plus second times derivative of first' u = x² (so du/dx = 2x) and v = sin x (so dv/dx = cos x). Then dy/dx = x² cos x + 2x sin x.
一个简单的记忆方法是“第一个乘第二个的导数加上第二个乘第一个的导数”。对于 y = x² sin x，令 u = x² (du/dx = 2x)，v = sin x (dv/dx = cos x)，则 dy/dx = x² cos x + 2x sin x。
The product rule is essential when the two functions cannot be easily multiplied out, and it often appears alongside the chain rule in exam questions.
当两个函数无法轻易乘开时，乘积法则就至关重要，考试中它经常与链式法则同时出现。
6. The Quotient Rule | 商法则
For a function of the form y = u(x) / v(x), where v(x) ≠ 0, the derivative is given by:
对于形如 y = u(x) / v(x) 的函数（其中 v(x) ≠ 0），其导数公式为：
dy/dx = (v du/dx − u dv/dx) / v²
The order matters: ' es derivative of the top minus top times derivative of the bottom, all over bottom squared' (du/dx = 1) and v = x+1 (dv/dx = 1). Then dy/dx = [(x+1)(1) − x(1)] / (x+1)² = 1 / (x+1)².
顺序很重要：“分母乘分子的导数减去分子乘分母的导数，再除以分母的平方”。对于 y = x / (x+1)，令 u = x (du/dx = 1)，v = x+1 (dv/dx = 1)，则 dy/dx = [(x+1)(1) − x(1)] / (x+1)² = 1 / (x+1)²。
Some students prefer to rewrite quotients as products using negative indices and then apply the product rule, which is a valid alternative. However, the quotient rule is often faster for rational functions.
有些学生更愿意利用负指数将商式改写为乘积，然后应用乘积法则，这也是可行的替代方法。但对于有理函数，使用商法则通常更快。
7. Differentiating Exponential and Logarithmic Functions | 指数函数和对数函数的微分
The exponential function eˣ is unique in that its derivative is itself. More generally, if y = ekx, where k is a constant, then dy/dx = k ekx.
指数函数 eˣ 的独特之处在于它的导数就是它本身。更一般地，若 y = ekx，其中 k 为常数，则 dy/dx = k ekx。
gument is a linear function, the chain rule gives: if y = ln(ax+b), then dy/dx = a/(ax+b).
对于自然对数，ln x 的导数为 1/x（x > 0）。当参数为线性函数时，结合链式法则可得：若 y = ln(ax+b)，则 dy/dx = a/(ax+b)。
Differentiating aˣ (where a > 0) requires rewriting as ex ln a. Then d/dx (aˣ) = aˣ n for modelling exponential growth and decay, which appear frequently in applied contexts.
对 aˣ（a > 0）求导需要改写为 ex ln a，从而 d/dx (aˣ) = aˣ ln a。这些结论构成了指数增长与衰减建模的基础，在应用题中经常出现。
8. Differentiating Trigonometric Functions | 三角函数的微分
The derivatives of the six trigonometric functions are essential knowledge. Starting from the basic sine and cosine:
六个三角函数的导数是必备知识。从基本的正弦和余弦开始：
d/dx (sin x) = cos x
d/dx (sin x) = cos x
d/dx (cos x) = −sin x
d/dx (cos x) = −sin x
sec² x. These results are standard and must be memorised, but they also can be derived from first principles if required.
利用商法则或将 tan x = sin x / cos x 进行改写，可得 d/dx (tan x) = sec² x。这些都是标准结果，需要记住，但必要时也可以由第一原理推导出来。
cot x, the derivatives are provided in the formula booklet, but understanding their derivations helps avoid sign errors.
当角度为线性函数时，结合链式法则可得：d/dx (sin(ax+b)) = a cos(ax+b)。对于 sec x、cosec x 和 cot x，公式表会提供其导数，但理解推导过程有助于避免符号错误。
9. Implicit Differentiation | 隐函数微分
When y is not explicitly expressed as a function of x, we can still differentiate by treating y as an implicit function. We differentiate both sides of the equation with respect to x, applying the chain rule to terms involving y, which introduces dy/dx.
当 y 没有明确表示为 x 的函数时，我们仍可将其视为隐函数进行微分。对等式两边关于 x 求导，对含有 y 的项应用链式法则，此时会引入 dy/dx。
is powerful for curves like circles and ellipses where isolating y is messy or impossible.
例如，已知 x² + y² = 25，逐项求导：2x + 2y (dy/dx) = 0，从而 dy/dx = −x/y。对于圆和椭圆这类难以将 y 单独分离出来的曲线，这一技巧非常有效。
Product and quotient rules still apply in implicit differentiation when terms involve products of x and y, such as x²y or sin(xy). Always remember to differentiate with respect to x and multiply each derivative of a y-term by dy/dx.
当出现 x 与 y 的乘积项（如 x²y 或 sin(xy)）时，隐函数微分中仍需使用乘积法则和商法则。务必记住始终对 x 求导，并将每个 y 相关项的导数乘以 dy/dx。
10. Higher-Order Derivatives | 高阶导数
The first derivative f '(x) represents the rate of change of f. Differentiating again gives the second derivative, f ''(x) or d²y/dx², which measures the rate of change of the gradient, i.e., the curvature or concavity of the function.
一阶导数 f '(x) 表示 f 的变化率。再次求导得到二阶导数 f ''(x) 或 d²y/dx²，它衡量梯度的变化率，即函数的曲率或凹凸性。
ives can be found for any differentiable function, but A-Level typically stops at the second or third derivative.
若 s(t) 是位移函数，则速度 v = ds/dt，加速度 a = d²s/dt²。任何可微函数都可继续求高阶导数，但 A-Level 通常最多只要求到二阶或三阶导数。
In questions involving stationary points, the second derivative test uses f ''(x): if f ''(a) > 0, the point is a local minimum; if f ''(a) < 0, it is a local maximum. When f ''(a) = 0, the test is inconclusive, and the first derivative test must be used.
在涉及驻点的问题中，二阶导数检验会用到 f ''(x)：若 f ''(a) > 0，该点为局部极小值；若 f ''(a) < 0，则为局部极大值。当 f ''(a) = 0 时，该方法无法得出结论，此时需用一阶导数检验。
11. Applications: Tangents and Normals | 应用：切线与法线
derivative is found, the equation of the tangent line at a point (a, f(a)) is y − f(a) = f '(a)(x − a). The normal is perpendicular to the tangent, so its gradient is −1 / f '(a), provided f '(a) ≠ 0.
一旦求得导数，曲线在点 (a, f(a)) 处的切线方程即为 y − f(a) = f '(a)(x − a)。法线与切线垂直，因此其斜率为 −1 / f '(a)（假设 f '(a) ≠ 0）。
For example, for the curve y = x³ − 3x at x = 2, f(2) = 2 and f ' − 2 = 9(x − 2), or y = 9x − 16. The normal gradient is −1/9, giving y − 2 = −1/9 (x − 2).
例如，对于曲线 y = x³ − 3x 在 x = 2 处，f(2) = 2，f '(2) = 3(2)² − 3 = 9。切线方程为 y − 2 = 9(x − 2)，即 y = 9x − 16。法线斜率为 −1/9，方程为 y − 2 = −1/9 (x − 2)。
12. Stationary Points and Optimisation | 驻点与最优化
Stationary points occur where the gradient is zero, i.e., f ' or local maxima, local minima, or points of inflection. Determining the nature of a stationary point is a core skill.
梯度为零处，即 f '(x) = 0，会出现驻点。这些点可能是局部极大值、局部极小值或拐点。判断驻点性质是一项核心技能。
Use the second derivative test, or examine the sign change of f '(x) around the point. For a maximum, f '(x) changes from positive to negative; for a minimum, from negative to positive. If the sign does not change, it is a point of inflection.
可采用二阶导数检验，或考察 f '(x) 在驻点附近的符号变化。对于极大值，f '(x) 由正变负；对于极小值，由负变正。如果符号不变，则为拐点。
Optimisation problems involve modelling a real-world situation with a function, finding its stationary points, and justifying the maximum or minimum value within a given domain. Typical contexts include minimising surface area for a fixed volume or maximising profit.
最优化问题需要先用函数对现实情景建模，然后求驻点，并在给定范围内论证最大值或最小值。典型情境包括在固定体积下最小化表面积，或最大化利润。
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