Differentiation | 微分考点精讲

📚 Differentiation | 微分考点精讲

Differentiation is the cornerstone of calculus, allowing us to describe how one quantity changes with respect to another. In OCR A-Level Mathematics, differentiation is tested across pure, mechanics, and even statistics contexts, from finding gradients of curves to solving optimisation problems. This guide covers the key concepts, techniques, and exam tips you need to master differentiation.

微积分是微积分的基石，让我们能够描述一个量相对于另一个量的变化率。在 OCR A-Level 数学考试中，微分贯穿纯数、力学甚至统计学，从求曲线斜率到解决最优化问题。本指南涵盖你需要掌握的核心概念、技巧和应试要点。

1. Introduction to Differentiation | 微分导论

Differentiation is the process of finding the derivative, f'(x) or dy/dx, which represents the instantaneous rate of change of a function y = f(x). Geometrically, the derivative at a point gives the gradient of the tangent to the curve at that point. The notation dy/dx emphasises the ratio of small changes in y and x, while f'(x) is a convenient shorthand introduced by Lagrange.

微分是求导数的过程，记作 f'(x) 或 dy/dx，它表示函数 y = f(x) 的瞬时变化率。从几何上看，某点的导数就是该点处曲线切线的斜率。符号 dy/dx 强调了 y 和 x 的微小变化之比，而 f'(x) 是拉格朗日引入的简便写法。

Leibniz notation: dy/dx – useful for chain rule and related rates / 莱布尼茨符号 dy/dx – 在链式法则和相关变化率中非常实用
Lagrange notation: f'(x) – compact, ideal for functions / 拉格朗日符号 f'(x) – 简洁，适合函数表示
Function of a function: If y = g(h(x)), then y depends on x through an intermediate variable / 复合函数: 若 y = g(h(x))，则 y 通过中间变量依赖于 x

2. Differentiation from First Principles | 从第一性原理求导

The formal definition of the derivative is the limit of the difference quotient: f'(x) = lim (h→0) [f(x+h) – f(x)] / h. This first-principles approach is the foundation of all differentiation rules. For a simple function like f(x) = x², we expand (x+h)², simplify, and let h approach zero to obtain f'(x) = 2x.

导数的正式定义是差分商的极限：f'(x) = lim(h→0) [f(x+h) – f(x)] / h。这种第一性原理方法是所有求导法则的基础。例如对 f(x) = x²，展开 (x+h)²，化简后令 h 趋于零，即得 f'(x) = 2x。

f'(x) = lim (h→0) (f(x+h) – f(x)) / h

You may be asked to prove the derivative of a quadratic or a simple rational function from first principles in the exam. Always write the limit explicitly, expand carefully, cancel h, and then evaluate the limit. This demonstrates a rigorous understanding of the gradient of a chord becoming the gradient of a tangent.

考试中可能要求从第一性原理证明二次函数或简单有理函数的导数。务必明确写出极限，仔细展开，消去 h，然后求极限。这体现了对弦斜率趋近于切线斜率的严谨理解。

3. Power Rule and Basic Derivatives | 幂函数求导法则及基本导数

The power rule is the most frequently used differentiation tool: if f(x) = xⁿ, then f'(x) = n xⁿ⁻¹. This holds for any real constant n, including negative and fractional powers. For example, d/dx (x⁻²) = –2 x⁻³ and d/dx (√x) = d/dx (x^½) = ½ x^⁻½.

幂函数求导法则是最常用的微分工具：若 f(x) = xⁿ，则 f'(x) = n xⁿ⁻¹。这适用于任意实数常数 n，包括负指数和分数指数。例如 d/dx (x⁻²) = –2 x⁻³，以及 d/dx (√x) = d/dx (x^(1/2)) = (1/2) x^(-1/2)。

d/dx (xⁿ) = n xⁿ⁻¹

You must also memorise the derivatives of fundamental functions: the derivative of a constant is zero; the derivative of eˣ is eˣ; the derivative of ln x is 1/x; and the derivative of sin x is cos x. These appear constantly in combination with the differentiation rules that follow.

你还必须熟记基本函数的导数：常数的导数为零；eˣ 的导数是 eˣ；ln x 的导数是 1/x；sin x 的导数是 cos x。它们将频繁地与后续求导法则结合出现。

Function f(x)	Derivative f'(x)
c (constant)	0
xⁿ	n xⁿ⁻¹
eˣ	eˣ
ln x	1/x
sin x	cos x
cos x	– sin x
tan x	sec² x

4. Differentiation Rules: Product, Quotient, Chain | 求导法则：积、商、链式

Three core rules handle products, quotients, and compositions of functions. The product rule: if y = u v, then dy/dx = u dv/dx + v du/dx. The quotient rule: if y = u/v, then dy/dx = (v du/dx – u dv/dx) / v². The chain rule: if y = f(g(x)), let u = g(x), then dy/dx = dy/du × du/dx.

三大核心法则分别处理函数的乘积、商和复合。积法则：若 y = u v，则 dy/dx = u dv/dx + v du/dx。商法则：若 y = u/v，则 dy/dx = (v du/dx – u dv/dx) / v²。链式法则：若 y = f(g(x))，设 u = g(x)，则 dy/dx = dy/du × du/dx。

Product: d/dx (u v) = u’ v + u v’
Quotient: d/dx (u/v) = (u’ v – u v’) / v²
Chain: dy/dx = dy/du × du/dx

When applying the chain rule to functions like y = (2x³ – 5)⁴, set u = 2x³ – 5, so y = u⁴. Then dy/du = 4u³ and du/dx = 6x², giving dy/dx = 4(2x³ – 5)³ × 6x². Always express your final answer in terms of the original variable x, and simplify where possible.

对诸如 y = (2x³ – 5)⁴ 的函数应用链式法则时，设 u = 2x³ – 5，则 y = u⁴。于是 dy/du = 4u³，du/dx = 6x²，得到 dy/dx = 4(2x³ – 5)³ × 6x²。务必用原变量 x 表达最终结果，并尽可能化简。

5. Differentiating Trigonometric Functions | 三角函数的微分

Trigonometric differentiation extends the basic rules to sin, cos, tan, and their reciprocals. You must know that d/dx (sin kx) = k cos kx, d/dx (cos kx) = –k sin kx, and d/dx (tan kx) = k sec² kx, where k is a constant. The chain rule is essential here: differentiate the outer trig function, then multiply by the derivative of the inner linear function.

三角函数的微分将基本法则扩展到 sin、cos、tan 及其倒数。你必须知道 d/dx (sin kx) = k cos kx，d/dx (cos kx) = –k sin kx，以及 d/dx (tan kx) = k sec² kx，其中 k 为常数。链式法则在这里至关重要：先对外层三角函数求导，再乘以内层线性函数的导数。

d/dx (sin 3x) = 3 cos 3x
d/dx (cos(2x + 1)) = –2 sin(2x + 1)

For more complicated arguments like sin² x or cos(f(x)), rewrite as (sin x)² or use the chain rule systematically. Watch for mixed rules: y = x² sin x requires the product rule; y = sin x / x requires the quotient rule. The derivative of sec x, csc x, and cot x can be derived but are less common in OCR; focus on sin, cos, tan.

对于更复杂的参数，如 sin² x 或 cos(f(x))，可改写为 (sin x)² 或系统地使用链式法则。注意混合法则：y = x² sin x 需要积法则；y = sin x / x 需要商法则。sec x、csc x、cot x 的导数可以推导，但在 OCR 考试中较少出现，重点放在 sin、cos、tan。

6. Differentiating Exponential and Logarithmic Functions | 指数与对数函数的微分

The exponential function eˣ is unique because its derivative is itself. For e^(kx), the chain rule gives d/dx (e^(kx)) = k e^(kx). The natural logarithm ln x differentiates to 1/x. For ln(kx), use the chain rule or the property ln(kx) = ln k + ln x, so the derivative is still 1/x because ln k is constant.

指数函数 eˣ 的独特之处在于它的导数就是它本身。对于 e^(kx)，由链式法则得 d/dx (e^(kx)) = k e^(kx)。自然对数 ln x 的导数是 1/x。对 ln(kx)，可利用链式法则或性质 ln(kx) = ln k + ln x，因此导数仍为 1/x，因为 ln k 是常数。

d/dx (e^(ax)) = a e^(ax)
d/dx (ln(ax)) = 1/x

For other bases, such as aˣ, rewrite using e: aˣ = e^(x ln a), so d/dx (aˣ) = aˣ ln a. This is not always required in OCR but can be examined. Logarithmic differentiation, where you take ln of both sides, is a powerful trick for functions of the form y = xˣ or products of many terms, but it is an extension topic.

对于其他底数，如 aˣ，可利用 e 改写：aˣ = e^(x ln a)，因此 d/dx (aˣ) = aˣ ln a。这在 OCR 中不总考到，但可能出现。对数微分法（两边取 ln）是处理形如 y = xˣ 或多因式乘积的有力技巧，但属于拓展内容。

7. Implicit Differentiation | 隐函数微分

When an equation defines y implicitly rather than explicitly as y = f(x), you differentiate term by term with respect to x, applying the chain rule whenever you differentiate a function of y. For example, to differentiate y² with respect to x, you write d/dx (y²) = 2y dy/dx. After differentiating, rearrange to solve for dy/dx.

当方程以隐式而非显式 y = f(x) 的形式给出 y 时，你需要逐项对 x 求导，每当对 y 的函数求导时都要应用链式法则。例如，对 y² 关于 x 求导，应写成 d/dx (y²) = 2y dy/dx。求导后，整理求出 dy/dx。

Example: x² + y² = 25 → 2x + 2y dy/dx = 0 → dy/dx = –x/y

Implicit differentiation is essential for curves like circles, ellipses, or xy + sin y = x². It is also used in related rates problems. Be careful with product terms like x y: differentiate as a product, giving 1·y + x dy/dx.

隐函数微分对于圆、椭圆等曲线，或诸如 xy + sin y = x² 的方程至关重要。它也用于相关变化率问题。处理乘积项如 x y 时要小心：按积法则求导，得到 1·y + x dy/dx。

8. Parametric Differentiation | 参数方程的微分

When a curve is defined by parametric equations x = f(t), y = g(t), the derivative dy/dx is found using dy/dx = (dy/dt) / (dx/dt), provided dx/dt ≠ 0. This follows from the chain rule: dy/dt = dy/dx × dx/dt. The second derivative d²y/dx² can be found by differentiating dy/dx with respect to t and dividing by dx/dt.

当曲线由参数方程 x = f(t)，y = g(t) 定义时，可用 dy/dx = (dy/dt) / (dx/dt) 求导，前提是 dx/dt ≠ 0。这源自链式法则：dy/dt = dy/dx × dx/dt。二阶导数 d²y/dx² 可通过将 dy/dx 对 t 求导再除以 dx/dt 得到。

dy/dx = (dy/dt) ÷ (dx/dt)
d²y/dx² = d/dt (dy/dx) ÷ (dx/dt)

Parametric differentiation is frequently tested with trigonometric parametrics, like x = a cos t, y = b sin t for an ellipse. In such cases, dy/dx = (b cos t) / (–a sin t) = –(b/a) cot t. Always simplify the final expression, and remember that the derivative is expressed in terms of the parameter t, not x or y.

参数方程微分常与三角函数参数一起考查，例如椭圆的参数方程 x = a cos t，y = b sin t。此时 dy/dx = (b cos t) / (–a sin t) = –(b/a) cot t。最终表达式务必化简，并记住导数用参数 t 表示，而非 x 或 y。

9. Second and Higher Order Derivatives | 二阶与高阶导数

The second derivative, denoted f”(x) or d²y/dx², is the derivative of the first derivative. It measures the rate of change of the gradient, which tells us about the curvature (concavity) of the graph. If f”(x) > 0, the graph is convex (curving upwards); if f”(x) < 0, it is concave (curving downwards).

二阶导数记作 f”(x) 或 d²y/dx²，是一阶导数的导数。它衡量斜率的变化率，从而揭示函数图像的凹凸性。若 f”(x) > 0，图像上凸（向上弯曲）；若 f”(x) < 0，图像下凹（向下弯曲）。

Concavity: f”(x) > 0 convex (∪), f”(x) < 0 concave (∩)

To find higher derivatives, simply differentiate repeatedly. For a polynomial, each differentiation reduces the degree by 1. The third derivative is occasionally used in mechanics (jerk) but rarely in pure OCR. The second derivative test for stationary points is a key application: if f”(x) > 0 at a stationary point, it is a local minimum; if f”(x) < 0, it is a local maximum.

求高阶导数只需反复求导即可。对于多项式，每次求导次数减少 1。三阶导数偶尔在力学（加加速度）中出现，但在 OCR 纯数中很少。二阶导数判别驻点的性质是一个关键应用：若驻点处 f”(x) > 0，则为局部极小值；若 f”(x) < 0，则为局部极大值。

10. Tangents and Normals | 切线与法线

Given a point (x₁, y₁) on a curve y = f(x), the gradient of the tangent is m = f'(x₁). The equation of the tangent is then y – y₁ = m (x – x₁). The normal is perpendicular to the tangent, so its gradient is –1/m (provided m ≠ 0). The corresponding normal equation is y – y₁ = (–1/m)(x – x₁).

给定曲线 y = f(x) 上一点 (x₁, y₁)，切线的斜率为 m = f'(x₁)。于是切线方程为 y – y₁ = m (x – x₁)。法线垂直于切线，因此其斜率为 –1/m（假设 m ≠ 0）。相应的法线方程为 y – y₁ = (–1/m)(x – x₁)。

Tangent: y – y₁ = m (x – x₁)
Normal: y – y₁ = –(1/m)(x – x₁)

OCR often asks for the tangent or normal at a specific point, which requires substituting x into the derivative and simplifying the equation. For curves defined implicitly or parametrically, find the gradient first, then proceed as above. Be careful with vertical tangents (infinite gradient) where normal is horizontal, and vice versa.

OCR 常要求求某特定点处的切线或法线，这需要将 x 代入导数并化简方程。对于隐式或参数方程定义的曲线，先求出斜率，再按上述步骤处理。要注意竖直切线（斜率无穷大）的情况，此时法线为水平线，反之亦然。

11. Stationary Points and Nature | 驻点及性质

Stationary points occur where dy/dx = 0. The nature (maximum, minimum, or point of inflection) can be determined by the second derivative test or by examining the sign change of dy/dx on either side. For a stationary point at x = a: if f”(a) > 0, it is a minimum; if f”(a) < 0, it is a maximum. If f''(a) = 0, the test is inconclusive; use the first derivative sign test instead.

驻点发生在 dy/dx = 0 的位置。其性质（极大值、极小值或拐点）可通过二阶导数判别法或检查 dy/dx 在两侧的符号变化来确定。对于 x = a 处的驻点：若 f”(a) > 0，则为极小值；若 f”(a) < 0，则为极大值。如果 f''(a) = 0，判别法失效，应改用一阶导数符号法。

Second derivative test: f”(a) > 0 ⇒ min; f”(a) < 0 ⇒ max

A stationary point of inflection occurs where the gradient is zero but the concavity changes, so f”(x) = 0 and changes sign. However, not every point where f”(x) = 0 is an inflection – you must confirm the sign change. OCR expects you to fully justify the nature using either method with clear working.

驻拐点发生在斜率为零但凹性改变处，即 f”(x) = 0 且符号改变。不过，并非所有 f”(x) = 0 的点都是拐点——你必须确认符号发生变化。OCR 要求你用任一方法完整证明驻点性质，并给出清晰步骤。

12. Real-World Applications | 实际应用

Differentiation models rates of change: velocity is the derivative of displacement with respect to time; acceleration is the second derivative. In optimisation problems, you find maximum or minimum values of a quantity (e.g. volume, area, cost) by setting the derivative to zero and verifying the nature. Related rates link two or more changing quantities, using the chain rule to connect their derivatives.

微分可以模拟变化率：速度是位移对时间的导数；加速度是位移的二阶导数。在最优化问题中，通过令导数为零并验证驻点性质，可求出某个量（如体积、面积、成本）的最大或最小值。相关变化率问题将两个或多个变化量联系起来，利用链式法则连接它们的导数。

Velocity v = ds/dt, Acceleration a = dv/dt = d²s/dt²

In OCR mechanics, differentiation is applied to motion in a straight line with variable acceleration. In pure contexts, you might minimise the surface area of a can for a fixed volume or find the maximum area of a rectangle inscribed in a curve. Always express the quantity to be optimised in terms of a single variable before differentiating, and clearly state the domain.

在 OCR 力学中，微分被应用于变加速直线运动。在纯数中，你可能需要最小化固定体积下罐子的表面积，或求内接于曲线的矩形的最大面积。务必在微分前将需优化的量表示为单一变量的函数，并清楚说明定义域。

Modelling tip: Draw a diagram, label variables, write a constraint equation, and then eliminate extra variables / 建模提示：画图，标注变量，列出约束方程，然后消去多余变量
Exam warning: If the second derivative test is messy, use the sign-change method for stationary points to save time / 考试提醒：若二阶导数判别法繁琐，可用符号变化法判断驻点以节省时间

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

Differentiation | 微分 考点精讲