📚 Mastering Differentiation for Edexcel A-Level Maths | 掌握 Edexcel A-Level 数学微分
Differentiation is one of the most heavily examined topics in Edexcel A-Level Mathematics. This revision guide covers the key techniques, from first principles to parametric differentiation, and shows how to apply them to gradients, stationary points, tangents, normals and optimisation problems. Work through each section systematically, make sure every formula is memorised, and practise with past-paper questions under timed conditions.
微分是 Edexcel A-Level 数学中考试频率最高的主题之一。本复习指南涵盖从第一性原理到参数方程微分的关键技巧，并展示如何将其应用于梯度、驻点、切线、法线和优化问题。请系统学习每一节，确保记住所有公式，并在计时条件下练习历年真题。
1. Differentiation from First Principles | 第一性原理微分
Differentiation from first principles defines the derivative as the limit of a gradient. For a function y = f(x), the derivative f'(x) is given by the limit of the difference quotient as h tends to 0.
第一性原理微分将导数定义为梯度的一个极限。对于函数 y = f(x)，导数 f'(x) 是差商在 h 趋于 0 时的极限。
f'(x) = lim(h → 0) [f(x + h) − f(x)] / h
This formula is essential for proving the derivative of basic powers such as x² or x³. In an exam, you may be asked to use first principles for a specific polynomial, so always expand f(x + h) fully and simplify before taking the limit.
该公式对于证明 x² 或 x³ 等基本幂函数的导数是必要的。在考试中，你可能会被要求对某个特定多项式使用第一性原理，因此务必先完整展开 f(x + h) 并化简，然后再取极限。
2. Basic Derivative Rules | 基本导数法则
The most fundamental rule is the power rule: to differentiate xⁿ, multiply by the power and reduce the power by one. This rule works for all real values of n, including negative and fractional powers.
最基本的法则是幂函数法则：对 xⁿ 求导时，乘以指数并将指数减一。该法则适用于所有实数值 n，包括负指数和分数指数。
d/dx (xⁿ) = n xⁿ⁻¹
You also need the constant multiple rule and the sum rule: constants can be taken outside the derivative, and the derivative of a sum is the sum of the derivatives. These rules allow you to differentiate any polynomial term by term.
你还需要掌握常数倍法则和加法法则：常数可以移到导数符号外，和的导数等于各项导数之和。这些法则让你能够对任何多项式逐项求导。
A quick reference table of common derivatives is useful for revision.
快速参考常见导数表对复习非常有用。
| f(x) | f'(x) |
| xⁿ | n xⁿ⁻¹ |
| k (constant) | 0 |
| kx | k |
| 1/x = x⁻¹ | −1/x² = −x⁻² |
| √x = x^(1/2) | 1/(2√x) = (1/2)x^(−1/2) |
Always rewrite expressions such as 1/x², √x or 3/x³ as powers of x before differentiating. Many errors occur when students try to apply the quotient rule unnecessarily or forget the negative sign.
在求导前，务必将 1/x²、√x 或 3/x³ 等表达式改写为 x 的幂。许多错误发生在学生不必要地使用商法则或忘记负号时。
3. Chain Rule | 链式法则
The chain rule is used to differentiate composite functions. If y is a function of u and u is a function of x, then the derivative of y with respect to x is the product of the derivative of y with respect to u and the derivative of u with respect to x.
链式法则用于对复合函数求导。如果 y 是 u 的函数，u 是 x 的函数，那么 y 对 x 的导数等于 y 对 u 的导数乘以 u 对 x 的导数。
dy/dx = dy/du × du/dx
For example, if y = (3x² + 5)⁴, set u = 3x² + 5. Then dy/du = 4u³ and du/dx = 6x, so dy/dx = 4(3x² + 5)³ × 6x = 24x(3x² + 5)³.
例如，如果 y = (3x² + 5)⁴，设 u = 3x² + 5。则 dy/du = 4u³，du/dx = 6x，因此 dy/dx = 4(3x² + 5)³ × 6x = 24x(3x² + 5)³。
In A-Level exams, the chain rule is often combined with other rules. When the inner function is linear, you can use the shortcut d/dx [f(ax + b)] = a f'(ax + b), but you must still show clear working for method marks.
在 A-Level 考试中，链式法则经常与其他法则结合使用。当内层函数为线性函数时，可以使用捷径 d/dx [f(ax + b)] = a f'(ax + b)，但你仍必须展示清晰的步骤以获得方法分。
4. Product and Quotient Rules | 乘积法则与商法则
The product rule is used when differentiating a function written as a product of two simpler functions. If u and v are both functions of x, then the derivative of uv is u times the derivative of v plus v times the derivative of u.
当函数写为两个较简单函数的乘积时，使用乘积法则。如果 u 和 v 都是 x 的函数，那么 uv 的导数等于 u 乘以 v 的导数加上 v 乘以 u 的导数。
d/dx (uv) = u dv/dx + v du/dx
For example, if y = x² sin x, set u = x² and v = sin x. Then dy/dx = x² cos x + 2x sin x. Factorising the result can sometimes earn further simplification marks.
例如，如果 y = x² sin x，设 u = x²，v = sin x。则 dy/dx = x² cos x + 2x sin x。对结果进行因式分解有时可以获得进一步的化简分。
The quotient rule applies when one function is divided by another. If y = u/v, then the derivative is v times du/dx minus u times dv/dx, all divided by v².
当一个函数除以另一个函数时，使用商法则。如果 y = u/v，那么导数等于 v 乘以 du/dx 减去 u 乘以 dv/dx，再整体除以 v²。
d/dx (u/v) = (v du/dx − u dv/dx) / v²
Remember to keep the order in the numerator correct: it is v du/dx first, then subtract u dv/dx. A common mistake is reversing the order or forgetting to square the denominator.
请记住分子中的顺序要正确：先写 v du/dx，再减去 u dv/dx。常见错误是颠倒顺序或忘记对分母平方。
5. Differentiating Exponentials and Logarithms | 指数函数与对数函数的微分
The exponential function eˣ is unique because its derivative is itself. More generally, if y = eᵏˣ, where k is a constant, then the derivative is k eᵏˣ.
指数函数 eˣ 的独特之处在于它的导数等于它本身。更一般地，如果 y = eᵏˣ，其中 k 为常数，那么导数为 k eᵏˣ。
d/dx (eˣ) = eˣ d/dx (eᵏˣ) = k eᵏˣ
The natural logarithm function ln x has derivative 1/x. For a linear argument, d/dx [ln(ax + b)] = a/(ax + b). This appears regularly in integration and differential equations.
自然对数函数 ln x 的导数为 1/x。对于线性自变量，d/dx [ln(ax + b)] = a/(ax + b)。这在积分和微分方程中经常出现。
d/dx (ln x) = 1/x d/dx [ln(ax + b)] = a/(ax + b)
You may also need to differentiate functions such as y = ln(3x² + 1) using the chain rule. Let u = 3x² + 1, so dy/dx = (1/u) × 6x = 6x/(3x² + 1).
你可能还需要使用链式法则对 y = ln(3x² + 1) 这类函数求导。令 u = 3x² + 1，则 dy/dx = (1/u) × 6x = 6x/(3x² + 1)。
6. Trigonometric Differentiation | 三角函数的微分
Edexcel A-Level requires you to memorise the standard derivatives of the six trigonometric functions. The three most important are sin x, cos x and tan x.
Edexcel A-Level 要求你记住六个三角函数的标准导数。其中最重要的三个是 sin x、cos x 和 tan x。
d/dx (sin x) = cos x
d/dx (cos x) = −sin x
d/dx (tan x) = sec² x
The derivatives of sec x, cosec x and cot x are less common but can appear in harder questions. These are d/dx (sec x) = sec x tan x, d/dx (cosec x) = −cosec x cot x and d/dx (cot x) = −cosec² x.
sec x、cosec x 和 cot x 的导数较少见，但可能出现在较难的题目中。这些导数分别是 d/dx (sec x) = sec x tan x、d/dx (cosec x) = −cosec x cot x 以及 d/dx (cot x) = −cosec² x。
When the angle is not simply x, you must use the chain rule. For example, d/dx [sin(2x + 1)] = 2 cos(2x + 1) and d/dx [tan 3x] = 3 sec² 3x.
当角度不完全是 x 时，你必须使用链式法则。例如，d/dx [sin(2x + 1)] = 2 cos(2x + 1)，d/dx [tan 3x] = 3 sec² 3x。
7. Implicit Differentiation | 隐函数微分
Implicit differentiation is used when y is not written explicitly as a function of x, for example in an equation such as x² + y² = 25. Differentiate both sides with respect to x, treating y as a function of x and using the chain rule.
当 y 没有明确写成 x 的函数时，例如在 x² + y² = 25 这样的方程中，需要使用隐函数微分。对等式两边关于 x 求导，将 y 视为 x 的函数并使用链式法则。
For x² + y² = 25, differentiating gives 2x + 2y dy/dx = 0. Rearranging yields dy/dx = −x/y. This expresses the gradient in terms of both x and y, which is perfectly acceptable.
对于 x² + y² = 25，求导得到 2x + 2y dy/dx = 0。整理后得到 dy/dx = −x/y。该式用 x 和 y 同时表示梯度，这是完全可以接受的。
In exam questions, you may then be asked to find the gradient at a specific point, or to find points where the tangent is horizontal or vertical. Always substitute the coordinates only after you have found an expression for dy/dx.
在考试题中，你可能会被要求求某一点的梯度，或求切线水平或竖直的点。务必在求出 dy/dx 的表达式之后再代入坐标。
8. Parametric Differentiation | 参数方程微分
When a curve is defined by parametric equations x = f(t) and y = g(t), the gradient dy/dx is found by dividing dy/dt by dx/dt. This comes from the chain rule: dy/dx = dy/dt × dt/dx, where dt/dx = 1/(dx/dt).
当曲线由参数方程 x = f(t) 和 y = g(t) 定义时，梯度 dy/dx 通过 dy/dt 除以 dx/dt 来求得。这来源于链式法则：dy/dx = dy/dt × dt/dx，其中 dt/dx = 1/(dx/dt)。
dy/dx = (dy/dt) / (dx/dt)
For example, if x = t² + 1 and y = 2t³, then dx/dt = 2t and dy/dt = 6t². Hence dy/dx = 6t² / 2t = 3t, provided dx/dt ≠ 0.
例如，如果 x = t² + 1 且 y = 2t³，那么 dx/dt = 2t，dy/dt = 6t²。因此 dy/dx = 6t² / 2t = 3t，前提是 dx/dt ≠ 0。
Questions on parametric curves often ask for the equation of a tangent or normal. You must find the gradient in terms of t, substitute the value of t at the given point, and then use the standard line equation y − y₁ = m(x − x₁).
关于参数曲线的题目常常要求求切线或法线方程。你必须先求出用 t 表示的梯度，代入给定点的 t 值，然后使用标准直线方程 y − y₁ = m(x − x₁)。
9. Stationary Points and Optimisation | 驻点与优化问题
A stationary point occurs where the gradient dy/dx is equal to zero. Solving f'(x) = 0 gives the x-coordinates of turning points, and substituting back into y = f(x) gives the corresponding y-coordinates.
驻点出现在梯度 dy/dx 等于零的位置。解方程 f'(x) = 0 可以得到拐点的 x 坐标，代入 y = f(x) 可得到对应的 y 坐标。
To classify a stationary point as a maximum or minimum, use the second derivative test. If f”(x) > 0, the point is a local minimum; if f”(x) < 0, it is a local maximum. If f''(x) = 0, the test is inconclusive and you should check the sign of f'(x) on either side.
要将驻点划分为极大值或极小值，可以使用二阶导数检验。如果 f”(x) > 0，该点为局部极小值；如果 f”(x) < 0，该点为局部极大值。如果 f''(x) = 0，检验无结论，此时应当检查 f'(x) 在两边的符号。
Optimisation problems ask you to find the maximum or minimum value of a quantity such as area, volume or cost. Set up an expression in one variable, differentiate, set the derivative equal to zero, and justify that your answer is a maximum or minimum.
优化问题要求你求面积、体积或成本等数量的最大值或最小值。建立单变量表达式，对其求导，令导数等于零，并证明所得结果为极大值或极小值。
10. Tangents, Normals and Increasing/Decreasing Functions | 切线、法线与增减函数
The derivative at a point gives the gradient of the tangent to the curve at that point. The tangent line can be found using y − y₁ = m(x − x₁), where m = f'(x₁).
函数在某点的导数给出了曲线在该点切线的斜率。切线方程可以使用 y − y₁ = m(x − x₁) 来求，其中 m = f'(x₁)。
The normal is perpendicular to the tangent, so its gradient is the negative reciprocal of the tangent gradient. If the tangent gradient is m, the normal gradient is −1/m.
法线垂直于切线，因此其斜率为切线斜率的负倒数。如果切线斜率为 m，则法线斜率为 −1/m。
A function is increasing on an interval where f'(x) > 0 and decreasing where f'(x) < 0. This sign analysis is also useful for sketching curves and justifying the nature of stationary points.
函数在 f'(x) > 0 的区间上递增，在 f'(x) < 0 的区间上递减。这种符号分析对于绘制曲线图和证明驻点性质也很有用。
11. Second Derivatives and Concavity | 二阶导数与凹凸性
The second derivative f”(x) measures the rate of change of the gradient. It tells you whether a curve is concave up or concave down. If f”(x) > 0 on an interval, the curve is convex (concave up); if f”(x) < 0, it is concave down.
二阶导数 f”(x) 衡量梯度的变化率。它可以告诉你曲线是向上凹还是向下凹。如果在某个区间上 f”(x) > 0，曲线是凸的（向上凹）；如果 f”(x) < 0，则曲线是向下凹的。
A point of inflection is where the concavity changes sign. To find a possible inflection point, solve f”(x) = 0, but you must also check that f”(x) changes sign around that value.
拐点是凹凸性改变符号的位置。要求可能的拐点，可解 f”(x) = 0，但你还必须检查 f”(x) 在该值附近是否改变符号。
In some Edexcel questions, second derivatives are linked to acceleration in mechanics, where displacement x is differentiated twice to give acceleration d²x/dt². The same mathematical rules apply.
在一些 Edexcel 题目中，二阶导数与力学中的加速度相关，位移 x 对 t 求二阶导得到加速度 d²x/dt²。相同的数学规则同样适用。
12. Exam Strategy and Common Mistakes | 考试策略与常见错误
Before starting a differentiation question, identify which rule is required: basic power rule, chain rule, product rule, quotient rule, implicit or parametric. Writing the rule down first can help you avoid structural errors.
开始做微分题之前，先确定需要使用哪条法则：基本幂函数法则、链式法则、乘积法则、商法则、隐函数还是参数方程。先写出法则有助于避免结构性错误。
One of the most common mistakes is forgetting to multiply by the derivative of the inner function when using the chain rule. For example, d/dx [cos 5x] is −5 sin 5x, not just −sin 5x.
最常见的错误之一是在使用链式法则时忘记乘以内层函数的导数。例如，d/dx [cos 5x] 应为 −5 sin 5x，而不仅仅是 −sin 5x。
Another frequent error is misapplying the quotient rule by writing u dv/dx − v du/dx instead of v du/dx − u dv/dx. Always start with v du/dx, then subtract u dv/dx.
另一个常见错误是误用商法则，写成 u dv/dx − v du/dx，而正确应为 v du/dx − u dv/dx。一定要从 v du/dx 开始，再减去 u dv/dx。
Finally, always simplify your answer if the question asks for it, and show every step of method. Marks are awarded for correct differentiation, substitution and classification, so even if your final answer is wrong, clear working can still earn method marks.
最后，如果题目要求化简，一定要化简答案，并展示每一步方法。评分会给正确的微分、代入和分类步骤分，因此即使最终答案错误，清晰的步骤仍然可以获得方法分。
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