📚 Differentiation | 微分
Differentiation is a core topic in A‑Level Mathematics that enables us to find the instantaneous rate of change of a function. From gradients of curves to optimisation and motion, calculus underpins much of modern science and engineering. In the Edexcel specification, you are expected to master a range of techniques – from first principles, standard derivatives, chain, product, and quotient rules, through to implicit and parametric differentiation. You must also be comfortable applying differentiation to tangents, normals, stationary points, and connected rates of change.
微分是A‑Level数学的核心内容，它帮助我们求一个函数在某点的瞬时变化率。从曲线斜率到最优化和运动学，微积分是现代科学和工程的重要基础。在Edexcel考纲中，你需要掌握从第一性原理、标准导数公式，到链式法则、乘积法则、商法则，再到隐函数微分和参数微分的各种技巧，并且能够将微分应用于切线、法线、驻点及相关变化率等问题。
1. The Derivative from First Principles | 从第一性原理求导
The derivative of a function f(x) at a point is defined as the limit of the average rate of change as the interval approaches zero: f'(x) = lim(h → 0) [f(x + h) − f(x)] / h he foundation of differentiation. For example, using first principles on f(x) = x² gives f'(x) = 2x. While exam questions usually don' way, understanding the limit definition is essential for grasping why the techniques work.
函数 f(x) 在一点的导数定义为当区间趋于零时平均变化率的极限：f'(x) = lim(h → 0) [f(x + h) − f(x)] / h。这是微分的基础。例如，从第一性原理求 f(x) = x² 的导数可得 f'(x) = 2x。虽然考试中很少要求你用这种方法推导所有公式，但理解极限定义是掌握各种微分技巧的关键。
0. This process reinforces the idea that the derivative measures the slope of the tangent.
一个关键能力是能够将定义应用于简单的多项式。写出 f(x+h)，展开，减去 f(x)，消去 h，然后令 h → 0。这个过程强化了导数衡量切线斜率这一概念。
2. Basic Rules: Power, Constant, and Sum | 基本法则：幂函数、常数及和差法则
The power rule is the most frequently used tool: for f(x) = xⁿ, f'(x) = n xⁿ⁻¹, where n is a real number. This works for negative and fractional powers as well. The constant rule states that the derivative of a constant is 0, and the sum rule says you can differentiate term by term.
幂函数法则是使用最频繁的工具：对于 f(x) = xⁿ，f'(x) = n xⁿ⁻¹，其中 n 为任意实数，也适用于负指数和分数指数。常数法则表明常数的导数为 0，和差法则意味着可以逐项求导。
7 is 0. Hence dy/dx = 12x² − 10x + 2. Remember to rewrite roots and reciprocals as powers: √x = x1/2, 1/x² = x⁻².
例如，求 y = 4x³ − 5x² + 2x − 7 的导数。逐项求导：4x³ 的导数是 12x²，−5x² 的导数是 −10x，2x 的导数是 2，−7 的导数是 0。因此 dy/dx = 12x² − 10x + 2。记住把根式和倒数写成幂的形式：√x = x1/2，1/x² = x⁻²。
| Function 函数 | Derivative 导数 |
|---|---|
| xⁿ | n xⁿ⁻¹ |
| k (constant) k（常数） | 0 |
| a f(x) + b g(x) | a f'(x) + b g'(x) |
3. The Chain Rule | 链式法则
(du/dx). In Leibniz notation this is very natural; in function notation it becomes f(g(x))' = f'(g(x)) · g'(x).
当函数由外函数和内函数复合而成时，我们需要运用链式法则：若 y = f(u) 且 u = g(x)，则 dy/dx = (dy/du) × (du/dx)。莱布尼茨记号使这一关系非常直观；用函数记号则可写为 f(g(x))' = f'(g(x)) · g'(x)。
us dy/dx = 4u³ · 6x = 24x(3x² + 5)³. The chain rule is also essential for functions like e2x, sin(5x), ln(x²+1).
例如：求 y = (3x² + 5)⁴ 的导数。令 u = 3x² + 5，则 y = u⁴。于是 dy/du = 4u³，du/dx = 6x。所以 dy/dx = 4u³ · 6x = 24x(3x² + 5)³。链式法则对于 e2x、sin(5x)、ln(x²+1) 等函数也至关重要。
Practice recognising the "inner function" quickly – it helps to spot squares, brackets, or denominators. For harder problems you may need to combine chain rule with other rules.
练习快速识别“内函数”——能帮助你发现平方、括号或分母。对于更复杂的问题，你可能需要结合链式法则和其他法则。
4. The Product Rule | 乘积法则
For two functions u(x) and v(x), the product rule states: d/dx (u v) = u' v + u v'. This is best remembered as "derivative of the first times the second, plus the first times the derivative of the second".
对于两个函数 u(x) 和 v(x)，乘积法则为：d/dx (u v) = u' v + u v'。简便记法是“前导乘后不导加前不导乘后导”。
Example: differentiate y = x² sin x. Let u = x², v = sin x. Then u' = 2x, v' ifferentiable functions, and you can extend it to three or more by applying it repeatedly.
示例：求 y = x² sin x 的导数。令 u = x²，v = sin x。则 u' = 2x，v' = cos x。于是 dy/dx = 2x sin x + x² cos x。乘积法则可以处理任意两个可导函数的乘积，如果需要三个或更多因子，可重复应用该法则。
Be careful to use brackets when the product appears inside a larger expression. Also, don't forget to simplify; sometimes common factors can be taken out.
当乘积嵌套在更大的表达式中时，要注意使用括号。另外不要忘记化简，有时可以提取公因式。
5. The Quotient Rule | 商法则
When y = u / v, the quotient rule gives: dy/dx = (u' v − u v') / v². A common mnemonic is "low d-high minus high d-low, over low squared". This rule is essential for rational functions such as (x+1)/(x²+2).
当 y = u / v 时，商法则为：dy/dx = (u' v − u v') / v²。常见的口诀是“分母导分子乘分母减分子导分母乘分母，除以分母的平方”。这个法则是处理形如 (x+1)/(x²+2) 的有理函数的关键。
Example: differentiate y = x² / (3x+1). Let u = x², v = 3x+1. Then u' = 2x, v' ting: dy/dx = (2x(3x+1) − x²·3) / (3x+1)² = (6x²+2x − 3x²) / (3x+1)² = (3x²+2x) / (3x+1)².
示例：求 y = x² / (3x+1) 的导数。令 u = x²，v = 3x+1。则 u' = 2x，v' = 3。代入得：dy/dx = (2x(3x+1) − x²·3) / (3x+1)² = (6x²+2x − 3x²) / (3x+1)² = (3x²+2x) / (3x+1)²。
It is often beneficial to simplify the numerator after applying the rule. In some cases, rewriting the quotient as a product (using negative powers) can be easier, but be prepared to use the quotient rule directly when instructed or when simplification is not straightforward.
在应用法则后简化分子通常是有益的。在某些情况下，将商式改写为乘积形式（使用负指数）可能更简单，但也要做好直接使用商法则的准备，尤其是当题目明确要求或化简不易时。
6. Derivatives of Exponential and Logarithmic Functions | 指数函数与对数函数的导数
The derivative of eˣ is special: d/dx (eˣ) = eˣ. If the exponent is a function of x, the chain rule gives: d/dx (ekx) = k ekx the natural logarithm, d/dx (ln x) = 1/x (for x > 0). More generally, d/dx (ln(f(x))) = f'(x) / f(x).
eˣ 的导数非常特殊：d/dx (eˣ) = eˣ。如果指数是 x 的函数，使用链式法则可得：d/dx (ekx) = k ekx。对于自然对数，d/dx (ln x) = 1/x（x > 0）。更一般地，d/dx (ln(f(x))) = f'(x) / f(x)。
For example, y = e2x → dy/dx = 2 e2x ). These derivatives appear frequently in modelling growth and decay, as well as in integration later.
例如，y = e2x → dy/dx = 2 e2x；y = ln(5x²+1) → dy/dx = (10x) / (5x²+1)。这些导数在增长与衰减模型以及后续的积分中都频繁出现。
Also note that for aˣ (a > 0), the derivative is aˣ ln a. This can be derived by rewriting aˣ = ex ln a and using the chain rule.
还需注意，对于 aˣ（a > 0），导数是 aˣ ln a。这可以通过将 aˣ 改写为 ex ln a 并结合链式法则推导出来。
7. Derivatives of Trigonometric Functions | 三角函数的导数
In A‑ sin x; d/dx (tan x) = sec² x. With chain rule extensions: d/dx (sin(kx)) = k cos(kx), etc.
在A‑Level中，角度始终使用弧度制。关键的导数有：d/dx (sin x) = cos x；d/dx (cos x) = −sin x；d/dx (tan x) = sec² x。结合链式法则可得：d/dx (sin(kx)) = k cos(kx) 等。
. First term: derivative = −3 sin(3x). Second term uses product rule: u = x, v = tan x → u' = 1, v' = sec² x, so derivative = 1·tan x + x·sec² x. Hence dy/dx = −3 sin(3x) + tan x + x sec² x.
示例：求 y = cos(3x) + x tan x 的导数。第一项导数为 −3 sin(3x)。第二项使用乘积法则：u = x，v = tan x → u' = 1，v' = sec² x，因此导数为 tan x + x sec² x。所以 dy/dx = −3 sin(3x) + tan x + x sec² x。
ng trig expressions to make differentiation easier (e.g., tan x = sin x / cos x and use quotient rule).
你还应熟悉 sec x 的导数为 sec x tan x，csc x 的导数为 −csc x cot x，cot x 的导数为 −csc² x，尽管这些较不常见。通过重写三角函数表达式（例如 tan x = sin x / cos x 并使用商法则）可以使求导更简单。
8. Implicit Differentiation | 隐函数微分
When y is not given explicitly as a function of x, we differentiate both sides of an equation term by term, treating y as a function of x. Each time you differentiate a term involving y, multiply by dy/dx (chain rule). This technique is used for circles, ellipses, and many mixed polynomial equations.
当 y 未以显函数形式给出时，我们对等式两边逐项求导，将 y 视为 x 的函数。每次对含有 y 的项求导时，要乘以 dy/dx（链式法则）。此方法常用于圆、椭圆及多种混合多项式方程。
4), substitute to get −3/4. Implicit differentiation often leads to dy/dx expressed in terms of both x and y.
示例：已知 x² + y² = 25，两边求导：2x + 2y (dy/dx) = 0 → dy/dx = −x/y。要求点 (3,4) 处的斜率，代入得 −3/4。隐函数微分常将 dy/dx 表示为 x 和 y 的函数。
When dealing with products like x²y, use the product rule on x²·y, giving 2x·y + x²·(dy/dx). Always remember to collect all dy/dx terms on one side and factorise.
在处理像 x²y 这样的乘积时，对 x²·y 使用乘积法则，得到 2x·y + x²·(dy/dx)。务必记住将所有含 dy/dx 的项移到等号一侧并提取公因式。
9. Parametric Differentiation | 参数微分
When x and y are both given in terms of a third variable (the parameter, often t), we find dy/dx using dy/dx = (dy/dt) / (dx/dt). This elegantly avoids having to eliminate the parameter.
当 x 和 y 均以第三个变量（参数，通常用 t 表示）给出时，我们用 dy/dx = (dy/dt) / (dx/dt) 求导。这样可以巧妙避免消去参数。
nt at a specific point, first find the corresponding t value, then evaluate dy/dx, and finally plug into the straight‑line equation.
示例：x = t² + 1，y = 2t³ − 3。则 dx/dt = 2t，dy/dt = 6t²。因此 dy/dx = (6t²) / (2t) = 3t（t ≠ 0）。要找到特定点的切线，先找出对应的参数 t 值，再计算 dy/dx，最后代入直线方程。
= [d/dt (dy/dx)] / (dx/dt). This allows you to determine concavity for parametric curves.
参数形式下的二阶导数公式为 d²y/dx² = d/dx (dy/dx) = [d/dt (dy/dx)] / (dx/dt)。这使你能判断参数曲线的凹凸性。
10. Second Derivatives and Concavity | 二阶导数与凹凸性
The second derivative, f''(x) or d²y/dx², is the derivative of the first derivative. It tells us how the gradient itself is changing. If f''(x) > 0 on an interval, the graph is concave up (or convex, depending on convention; in Edexcel we say "convex" and "concave"). If f''(x) < 0, the graph is concave down.
二阶导数 f''(x) 或 d²y/dx² 是一阶导数的导数。它告诉我们斜率本身是如何变化的。如果在某区间上 f''(x) > 0，则图像向上凹（在Edexcel术语中称为凸函数）；若 f''(x) < 0，则图像向下凹。
nts where the concavity changes are called points of inflection. At an inflection point, f'' derivative must change sign. Identifying these helps in sketching curves and understanding the behaviour of graphs.
凹凸性发生改变的点称为拐点。在拐点处，f''(x) = 0 或不存在，且二阶导数必须变号。识别拐点有助于描绘曲线和了解图像的整体走势。
Example: y = x³ − 3x → y' = 3x² − 3, y'' = 6x. Setting y'' = 0 gives x = 0. Check sign on either side: for x < 0, y'' < 0, concave; for x > 0, y'' > 0, convex. Thus (0,0) is a point of inflection.
示例：y = x³ − 3x → y' = 3x² − 3，y'' = 6x。令 y'' = 0 得 x = 0。检查两侧符号：x < 0 时 y'' < 0，向下凹；x > 0 时 y'' > 0，向上凹。因此 (0,0) 为拐点。
11. Stationary Points and Optimization | 驻点与最优化
oints occur where dy/dx = 0. To classify them, we can use the second derivative test: if f''(x) > 0 it's a local minimum; if f''(x) < 0, a local maximum; if f''(x) = 0 the test is inconclusive and we must examine the sign of dy/dx either side.
驻点出现在 dy/dx = 0 的地方。对它们进行分类，我们可以采用二阶导数判别法：若 f''(x) > 0 则为局部极小值；若 f''(x) < 0 则为局部极大值；若 f''(x) = 0 则该方法无效，需检查 dy/dx 左右两侧的符号。
Example: find and classify stationary points of f(x) = 2x³ − 9x² + 12x. f'(x) = 6x² − 18x + 12 = 6(x² − 3x + 2) = 6(x−1)(x−2). Stationary points at x=1 and x=2. f''(x) = 12x − 18. At x=1, f''(1) = −6 < 0 so maximum; at x=2, f''(2) = 6 > 0 so minimum. Then find y‑coordinates.
示例：求 f(x) = 2x³ − 9x² + 12x 的驻点并分类。f'(x) = 6x² − 18x + 12 = 6(x−1)(x−2)。驻点在 x=1 和 x=2。f''(x) = 12x − 18。x=1 时 f''(1) = −6 < 0 故为极大值；x=2 时 f''(2) = 6 > 0 故为极小值。接着求出相应的 y 坐标。
Optimisation problems involve using stationary points to maximise or minimise quantities such as area, volume, or cost. Always confirm the nature of the point and check endpoints if the domain is restricted.
最优化问题利用驻点来最大化或最小化面积、体积或成本等量。务必确认驻点的性质，并在定义域有界时检查端点。
12. Applications: Tangents, Normals, and Rates of Change | 应用：切线、法线与变化率
pendicular to the tangent, so its gradient is −1/m (provided m ≠ 0), and its equation can be written similarly.
导数给出某点处切线的斜率。切线方程为 y − y₁ = m (x − x₁)，其中 m = dy/dx 在 (x₁, y₁) 处的值。法线与切线垂直，因此其斜率为 −1/m（若 m ≠ 0），方程形式类似。
ses at a certain rate, the rate of increase of volume is found by dV/dt = (dV/dr) × (dr/dt). This is often called "connected rates of change".
变化率问题涉及用链式法则联系两个或多个导数。例如，若球形气球的半径以某速率增大，体积的增大速率可通过 dV/dt = (dV/dr) × (dr/dt) 求得。这常被称为“相关变化率”。
Always write down the given rates and the required rate, identify the linking formula, differentiate it implicitly with respect to time t, and substitute known values. Don't be tempted to plug numbers in before differentiating.
始终写下已知的变化率与要求的变化率，找出联系公式，对时间 t 隐式求导，再代入已知值。在求导之前不要急于代入数字。
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