📚 PDF Joiner (4) 058: Differentiation Techniques for Edexcel A-Level Maths | PDF合并文档(4)第58页：Edexcel A-Level 数学微分技巧
In this Edexcel A-Level Mathematics revision guide, we focus on differentiation, one of the most heavily examined topics in the Pure Mathematics papers. The techniques covered here include first principles, the chain, product and quotient rules, trigonometric and exponential functions, implicit and parametric differentiation, stationary points and connected rates of change.
在这份 Edexcel A-Level 数学复习指南中，我们重点讲解纯数学试卷中考查频率最高的主题之一——微分。本文涵盖的技术包括第一原理、链式法则、乘法法则、除法法则、三角函数与指数函数、隐函数微分、参数方程微分、驻点以及相关变化率。
1. First Principles and the Definition of the Derivative | 第一原理与导数定义
The derivative of a function f(x) at a point x is defined as the limit of the average rate of change as h approaches 0.
函数 f(x) 在 x 处的导数定义为当 h 趋近于 0 时平均变化率的极限。
f ‘(x) = lim (h → 0) [f(x + h) – f(x)] / h
This limit, when it exists, gives the slope of the tangent to the curve y = f(x) at that point. If the limit does not exist, the function is not differentiable there.
当该极限存在时，它给出曲线 y = f(x) 在该点切线的斜率。如果极限不存在，则函数在该点不可微。
For example, to differentiate f(x) = x² from first principles, substitute into the definition and simplify.
例如，要从第一原理对 f(x) = x² 求导，需要代入定义并化简。
f ‘(x) = lim (h → 0) [(x + h)² – x²] / h = lim (h → 0) (2x h + h²) / h = 2x
2. Basic Differentiation Rules and the Power Rule | 基本微分法则与幂法则
The power rule is the foundation of most differentiation in A-Level Mathematics. For any real constant n, the derivative of xⁿ is n xⁿ⁻¹.
幂法则是 A-Level 数学中大多数微分运算的基础。对于任意实常数 n，xⁿ 的导数为 n xⁿ⁻¹。
d/dx (xⁿ) = n xⁿ⁻¹
The derivative of a constant is zero, and differentiation is linear, so constants can be taken outside the derivative and sums can be differentiated term by term.
常数的导数为零，微分运算是线性的，因此常数可以提到导数符号外，和式可以逐项求导。
d/dx [a f(x) + b g(x)] = a f ‘(x) + b g ‘(x)
| Function f(x) | 函数 | Derivative f ‘(x) | 导数 |
|---|---|
| x² | 2x |
| x³ | 3x² |
| x⁻¹ | -x⁻² |
| √x = x^(1/2) | (1/2) x^(-1/2) |
3. The Chain Rule | 链式法则
The chain rule is used when differentiating a composite function y = f(g(x)). Let u = g(x) and y = f(u); then the derivative is the product of the derivatives with respect to each variable.
链式法则用于对复合函数 y = f(g(x)) 求导。设 u = g(x)，y = f(u)，则导数是对每个变量求导的乘积。
dy/dx = dy/du × du/dx
For example, if y = (3x² + 5)⁴, set u = 3x² + 5. Then y = u⁴, dy/du = 4u³ and du/dx = 6x, so the derivative is:
例如，如果 y = (3x² + 5)⁴，设 u = 3x² + 5。那么 y = u⁴，dy/du = 4u³，du/dx = 6x，因此导数为：
dy/dx = 4u³ × 6x = 24x(3x² + 5)³
In applied problems, the chain rule often appears when one variable depends on another through an intermediate variable.
在应用问题中，链式法则经常出现在一个变量通过中间变量依赖于另一个变量的情形中。
4. The Product Rule | 乘法法则
When differentiating a product of two functions y = u(x) v(x), the product rule must be used. It states that the derivative is the first function times the derivative of the second plus the second function times the derivative of the first.
当对两个函数的乘积 y = u(x) v(x) 求导时，必须使用乘法法则。它表明导数等于第一个函数乘以第二个函数的导数，再加上第二个函数乘以第一个函数的导数。
d/dx (u v) = u dv/dx + v du/dx
For example, for y = x² sin x, let u = x² and v = sin x. Then u’ = 2x and v’ = cos x, so the derivative is:
例如，对于 y = x² sin x，设 u = x²，v = sin x。则 u’ = 2x，v’ = cos x，因此导数为：
dy/dx = x² cos x + 2x sin x
It is important to recognise when a product is present; otherwise, applying the power rule directly will give an incorrect result.
识别是否存在乘积非常重要；否则直接套用幂法则会得到错误结果。
5. The Quotient Rule | 除法法则
For a quotient y = u(x) / v(x), provided v(x) ≠ 0, the derivative is given by the quotient rule. The numerator is the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all divided by the square of the denominator.
对于商 y = u(x) / v(x)，在 v(x) ≠ 0 的前提下，导数由除法法则给出。分子为分母乘以分子的导数减去分子乘以分母的导数，整个结果再除以分母的平方。
d/dx (u/v) = (v du/dx – u dv/dx) / v²
For example, if y = x / (x² + 1), then u = x, v = x² + 1, u’ = 1 and v’ = 2x. Substituting gives:
例如，如果 y = x / (x² + 1)，则 u = x，v = x² + 1，u’ = 1，v’ = 2x。代入后得到：
dy/dx = [(x² + 1)(1) – x(2x)] / (x² + 1)² = (1 – x²) / (x² + 1)²
Many exam questions combine the quotient rule with the chain or product rule, so it is essential to recognise the order of operations.
许多考试题目会将除法法则与链式法则或乘法法则结合，因此识别运算顺序至关重要。
6. Differentiating Trigonometric Functions | 三角函数的微分
The standard derivatives of the three main trigonometric functions must be memorised. They are frequently used in both pure and applied questions.
三个主要三角函数的标准导数必须牢记。它们在纯数学和应用题中都会频繁使用。
d/dx (sin x) = cos x
d/dx (cos x) = -sin x
d/dx (tan x) = sec² x
When the angle is a linear function of x, such as ax + b, apply the chain rule and multiply by the coefficient a.
当角度是 x 的线性函数时，例如 ax + b，需要应用链式法则并乘以系数 a。
d/dx [sin(ax + b)] = a cos(ax + b)
For example, d/dx [cos(5x – 2)] = -5 sin(5x – 2). Always check whether the argument is a simple x or a more complex expression before differentiating.
例如，d/dx [cos(5x – 2)] = -5 sin(5x – 2)。在求导之前，一定要检查角度部分是简单的 x 还是更复杂的表达式。
7. Differentiating Exponential and Logarithmic Functions | 指数函数与对数函数的微分
The natural exponential function eˣ is unique because its derivative is itself. For a general exponential function with a linear exponent, multiply by the coefficient of the exponent.
自然指数函数 eˣ 的特殊之处在于它的导数就是它本身。对于指数为线性函数的一般指数函数，需要乘以指数中的系数。
d/dx (eˣ) = eˣ
d/dx (e^(kx)) = k e^(kx)
The natural logarithm differentiates to the reciprocal of x. For a composite argument, the chain rule gives the derivative of the inside divided by the inside.
自然对数的导数是 x 的倒数。对于复合参数，链式法则给出的结果是内层函数的导数除以内层函数本身。
d/dx (ln x) = 1/x
d/dx [ln(ax + b)] = a / (ax + b)
For a general exponential base a, the derivative is aˣ ln a. This is less common in Edexcel A-Level but still part of the specification.
对于一般指数底数 a，导数为 aˣ ln a。这在 Edexcel A-Level 中较少见，但仍属于考纲范围。
8. Implicit Differentiation | 隐函数微分
Implicit differentiation is used when y is not explicitly given as a function of x. Differentiate both sides of the equation with respect to x, treating y as a function of x and applying the chain rule to any y term.
当 y 没有显式地表示为 x 的函数时，使用隐函数微分。对方程两边关于 x 求导，将 y 视为 x 的函数，并对任何含有 y 的项应用链式法则。
For example, consider the circle equation x² + y² = 25. Differentiate both sides with respect to x:
例如，考虑圆的方程 x² + y² = 25。对方程两边关于 x 求导：
2x + 2y dy/dx = 0
Rearrange to solve for dy/dx, which gives the gradient of the tangent at any point on the circle.
重新整理并解出 dy/dx，即可得到圆上任意一点切线的斜率。
dy/dx = -x / y
Implicit differentiation is especially useful for curves such as ellipses, hyperbolas and more general relations where separating y is difficult or impossible.
隐函数微分对于椭圆、双曲线以及难以或无法分离出 y 的一般关系式尤为有用。
9. Parametric Differentiation | 参数方程微分
When a curve is defined by parametric equations x = f(t) and y = g(t), the derivative dy/dx is found by dividing dy/dt by dx/dt, provided dx/dt is not zero.
当曲线由参数方程 x = f(t) 和 y = g(t) 定义时，导数 dy/dx 等于 dy/dt 除以 dx/dt，前提是 dx/dt 不为零。
dy/dx = (dy/dt) / (dx/dt)
For example, if x = t² + 1 and y = 2t + 3, then dx/dt = 2t and dy/dt = 2, so:
例如，如果 x = t² + 1，y = 2t + 3，则 dx/dt = 2t，dy/dt = 2，因此：
dy/dx = 2 / (2t) = 1 / t
To find the equation of a tangent to a parametric curve, first find the value of t at the point, then compute dy/dx and use the point-slope form.
要求参数曲线上某点的切线方程，首先确定该点对应的 t 值，然后计算 dy/dx，并使用点斜式方程。
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