Differentiation | 微分

📚 Differentiation | 微分

The study of differentiation lies at the heart of A-Level Mathematics. It equips you with the tools to measure how one quantity changes in relation to another, forming the foundation for everything from curve sketching to real‑world optimisation problems. In the Edexcel specification, differentiation is a major pure mathematics topic that connects algebra, trigonometry, exponentials and logarithms.

微分的学是 A-Level 数学的核心。它让你掌握度量一个量相对于另一个量如何变化的工具，为从曲线绘制到现实世界最优化问题奠定了基。在 Edexcel 考试大纲中，微分是一个重要的纯数学主题，将代数、三角学、指数函数和对数函数连接起来。

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1. Introduction to the Derivative | 导数简介

The derivative of a function f(x) measures the instantaneous rate of change of f with respect to x. Geometrically, it gives the slope of the tangent line to the graph y = f(x) at any chosen point.

函数 f(x) 的导数度量了 f 关于 x 的瞬时变化率。在几何上，它给出了曲线 y = f(x) 在任意选定点处切线的斜率。

Common notations for the derivative include f'(x), dy/dx and d/dx[f(x)]. Understanding these symbols is essential before tackling the rules and applications that follow.

导数常用的记法包括 f'(x)、dy/dx 和 d/dx[f(x)]。在学后续的求导法则和应用之前，理解这些符号至关重要。

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2. Differentiation from First Principles | 第一原理求导

The derivative can be defined using the limit of a difference quotient: f'(x) = lim (h → 0) [f(x+h) − f(x)] / h. This process is called differentiation from first principles and links the algebraic idea of gradient to the formal definition of the derivative.

导数可以通过差商的极限来定义：f'(x) = lim (h → 0) [f(x+h) − f(x)] / h。这个过程被称为从第一原理求导，将斜率的代数概念与导数的正式定义联系起来。

For example, to differentiate f(x) = x² from first principles, one expands (x+h)² = x² + 2xh + h², subtracts x², and simplifies the quotient to 2x + h, which tends to 2x as h → 0.

例如，从第一原理求 f(x) = x² 的导数，先展开 (x+h)² = x² + 2xh + h²，减去 x²，然后将商简化为 2x + h，当 h → 0 时趋近于 2x。

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3. Basic Rules of Differentiation | 基本求导法则

Differentiation becomes efficient when we apply the power rule: if f(x) = xⁿ, then f'(x) = n xⁿ⁻¹. This rule, together with the constant multiple rule and the sum rule, makes it possible to differentiate any polynomial term by term.

当我们应用幂法则时，求导变得高效：如果 f(x) = xⁿ，则 f'(x) = n xⁿ⁻¹。该法则与常数倍法则以及和差法则一起，使我们能够对任意多项式逐项求导。

The constant multiple rule states that d/dx [c·f(x)] = c·f'(x), and the sum rule says d/dx [f(x) ± g(x)] = f'(x) ± g'(x). These simple building blocks underpin all the more advanced techniques.

常数倍法则指出 d/dx [c·f(x)] = c·f'(x)，和差法则指出 d/dx [f(x) ± g(x)] = f'(x) ± g'(x)。这些简单的基块支撑着所有更高级的技巧。

Power Rule: d/dx (xⁿ) = n xⁿ⁻¹

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4. Differentiating Exponential and Logarithmic Functions | 指数与对数函数求导

The derivative of eˣ is remarkably simple: d/dx (eˣ) = eˣ. This property makes the natural exponential function extremely important in calculus and in modelling growth and decay.

eˣ 的导数非常简单：d/dx (eˣ) = eˣ。这一性质使得自然指数函数在微积分以及增长和衰减建模中极其重要。

For the natural logarithm, d/dx (ln x) = 1/x, provided x > 0. When dealing with other bases, use d/dx (aˣ) = aˣ ln a and apply the chain rule for more complicated exponents or logarithms of inner functions.

对于自然对数，d/dx (ln x) = 1/x（要求 x > 0）。在处理其他底数时，使用 d/dx (aˣ) = aˣ ln a，并对于更复杂的指数或内函数的对数应用链式法则。

d/dx (eˣ) = eˣ,    d/dx (ln x) = 1/x,    d/dx (aˣ) = aˣ ln a

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5. Differentiating Trigonometric Functions | 三角函数的求导

In the Edexcel A-Level, you must know the derivatives of the six basic trigonometric functions. The most important ones are sin x, cos x and tan x, which appear regularly in calculus problems.

在 Edexcel A-Level 中，你必须掌握六个基本三角函数的导数。最重要的是 sin x、cos x 和 tan x，它们在微积分问题中经常出现。

The standard results are: d/dx (sin x) = cos x; d/dx (cos x) = −sin x; d/dx (tan x) = sec² x. These are derived from first principles or from the quotient of sin and cos.

标准的求导结果是：d/dx (sin x) = cos x；d/dx (cos x) = −sin x；d/dx (tan x) = sec² x。这些结果可以从第一原理或通过 sin 与 cos 的商推导出来。

For the reciprocal functions, the derivatives are d/dx (sec x) = sec x tan x, d/dx (cosec x) = −cosec x cot x, and d/dx (cot x) = −cosec² x. Memorising all six results saves time in the exam.

对于倒数函数，导数为 d/dx (sec x) = sec x tan x，d/dx (cosec x) = −cosec x cot x，以及 d/dx (cot x) = −cosec² x。记住所有六个结果能在考试中节省时间。

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6. The Product Rule | 乘积法则

When differentiating a product of two functions, you cannot simply multiply their individual derivatives. The product rule states: if y = u v, where u and v are functions of x, then dy/dx = u dv/dx + v du/dx.

当对两个函数的乘积求导时，不能简单地将各自的导数相乘。乘积法则指出：如果 y = u v，其中 u 和 v 都是 x 的函数，则 dy/dx = u dv/dx + v du/dx。

In function notation, this is equivalent to (fg)’ = f g’ + g f’. It is good practice to identify u and v clearly, find their derivatives separately, and then assemble the result using the formula.

用函数记法，这等价于 (fg)’ = f g’ + g f’。最佳做法是清晰地标出 u 和 v，分别求出它们的导数，然后用公式组合起最终结果。

d/dx (uv) = u dv/dx + v du/dx

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7. The Quotient Rule | 商法则

For a function expressed as a quotient, y = u/v, the quotient rule gives its derivative. The formula is dy/dx = (v du/dx − u dv/dx) / v². It is crucial to subtract the terms in the correct order to avoid sign errors.

对于表示为商的函数 y = u/v，商法则给出了它的导数。公式为 dy/dx = (v du/dx − u dv/dx) / v²。必须按正确的顺序减去项，以避免符号错误。

An equivalent version using function notation is (f/g)’ = (g f’ − f g’) / g². When applied to rational functions or to expressions like tan x = sin x / cos x, the quotient rule produces the derivatives you already know.

使用函数记法的等价形式是 (f/g)’ = (g f’ − f g’) / g²。当应用到有理函数或像 tan x = sin x / cos x 这样的表达式时，商法则可以得出你已经学过的导数结果。

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8. The Chain Rule | 链式法则

The chain rule is used to differentiate composite functions. If y = f(g(x)), or in other words y is a function of an inner function u = g(x), then dy/dx = dy/du × du/dx.

链式法则用于对复合函数求导。如果 y = f(g(x))，换句话说 y 是内函数 u = g(x) 的函数，那么 dy/dx = dy/du × du/dx。

In Lagrange notation, this reads [f(g(x))]’ = f'(g(x))·g'(x). The chain rule appears everywhere in A-Level calculus, especially when integrating by substitution or differentiating exponential, logarithmic and trigonometric functions with linear inner arguments.

用拉格朗日记法，这写作 [f(g(x))]’ = f'(g(x))·g'(x)。链式法则在 A-Level 微积分中无处不在，特别是在用换元法积分或对内函数为线性的指数、对数和三角函数求导时。

dy/dx = dy/du × du/dx

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9. Implicit Differentiation | 隐函数微分

When an equation defines y implicitly as a function of x (for example, x² + y² = 25), we use implicit differentiation. The method involves differentiating both sides of the equation with respect to x, treating y as a function of x and using the chain rule on any y‑terms.

当一个方程将 y 隐式地定义为 x 的函数（例如 x² + y² = 25）时，我们使用隐函数微分。该方法涉及对方程两边关于 x 求导，将 y 视为 x 的函数，并对包含 y 的项使用链式法则。

For the circle x² + y² = 25, differentiation gives 2x + 2y·dy/dx = 0, and solving yields dy/dx = −x/y. Implicit differentiation is essential for handling curves that cannot be written easily as y = f(x) and for related rates problems.

对于圆 x² + y² = 25，求导得到 2x + 2y·dy/dx = 0，解出 dy/dx = −x/y。隐函数微分对于处理难以写作 y = f(x) 的曲线以及相关变化率问题至关重要。

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10. Parametric Differentiation | 参数微分

When a curve is defined by parametric equations x = f(t) and y = g(t), the derivative dy/dx is found using dy/dx = (dy/dt) / (dx/dt), provided dx/dt ≠ 0.

当一条曲线由参数方程 x = f(t) 和 y = g(t) 定义时，使用 dy/dx = (dy/dt) / (dx/dt) 可求出导数 dy/dx，只要 dx/dt ≠ 0。

Second derivatives for parametric curves require a further application of the chain rule: d²y/dx² = d/dx (dy/dx) = [d/dt (dy/dx)] / (dx/dt). This often appears in questions involving tangents, normals and stationary points defined parametrically.

参数曲线的二阶导数需要进一步应用链式法则：d²y/dx² = d/dx (dy/dx) = [d/dt (dy/dx)] / (dx/dt)。这经常出现在涉及参数定义的切线、法线和驻点的问题中。

dy/dx = (dy/dt) ÷ (dx/dt)

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11. Second Derivatives and Concavity | 二阶导数与凹凸性

The second derivative, denoted f”(x) or d²y/dx², represents the rate of change of the gradient. It tells us about the concavity of a curve: if f”(x) > 0, the curve is convex (concave up); if f”(x) < 0, it is concave (concave down).

二阶导数记作 f”(x) 或 d²y/dx²，表示梯度的变化率。它告诉我们曲线的凹凸性：如果 f”(x) > 0，曲线是下凸的（开口向上）；如果 f”(x) < 0，曲线是上凸的（开口向下）。

A point where the concavity changes sign is called a point of inflection. At such a point, f”(x) = 0 is a necessary but not sufficient condition; you must check that the sign of f”(x) changes on either side.

凹凸性发生变化的点称为拐点。在这样的点处，f”(x) = 0 是必要条件但并不充分；必须检查在点的左右两侧 f”(x) 的符号确实发生了变化。

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12. Applications: Tangents, Normals, and Optimisation | 应用：切线、法线与最优化

The gradient of the tangent to a curve at x = a is given by f'(a). The equation of the tangent is then y − f(a) = f'(a)(x − a), while the normal has slope −1/f'(a) (provided the tangent is not horizontal).

曲线在 x = a 处的切线梯度由 f'(a) 给出。于是，切线方程为 y − f(a) = f'(a)(x − a)，而法线的斜率为 −1/f'(a)（假定切线不水平）。

Stationary points occur where f'(x) = 0. You can classify them using the second derivative test: f”(x) > 0 indicates a local minimum, f”(x) < 0 a local maximum. If f''(x) = 0, the test is inconclusive and you should examine the sign of f'(x) either side.

驻点出现在 f'(x) = 0 处。你可以用二阶导数检验对其进行分类：f”(x) > 0 表示局部极小值，f”(x) < 0 表示局部极大值。如果 f''(x) = 0，检验失效，此时应检查 f'(x) 在点两侧的符号。

Optimisation problems bring all these ideas together: you model a real‑world quantity as a function, differentiate to find critical points, and use the second derivative or boundary checks to determine maximum or minimum values.

最优化问题将所有这些思想融汇在一起：你将一个现实世界的量建模为一个函数，求导找到临界点，并利用二阶导数或边界检验来确定最大值或最小值。

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