📚 Finding the Derivative | 求导数
In A-Level Mathematics, differentiation is one of the most important tools in calculus. This article covers the key techniques for finding derivatives, from first principles to the chain rule, product rule and quotient rule, with Edexcel-style worked examples and exam tips.
在 A-Level 数学中，微分是微积分最重要的工具之一。本文涵盖求导数的关键方法，从第一原理到链式法则、乘积法则和商法则，并配有 Edexcel 风格的例题和考试技巧。
1. What is Differentiation? | 什么是微分？
Differentiation is the process of finding the derivative of a function. The derivative measures the instantaneous rate of change of one quantity with respect to another. Geometrically, for a curve y = f(x), the derivative at a point gives the gradient of the tangent to the curve at that point.
微分是求函数导数的过程。导数衡量一个量对另一个量的瞬时变化率。从几何上看，对于曲线 y = f(x)，某一点的导数给出了该点处切线的斜率。
The derivative is written in several ways: f'(x), dy/dx, or d/dx [f(x)]. Each notation means exactly the same thing. On an Edexcel paper, you must be able to switch between these notations confidently.
导数有多种写法：f'(x)、dy/dx 或 d/dx [f(x)]。每种记号含义完全相同。在 Edexcel 考试中，你必须能够熟练地在这些记号之间转换。
2. First Principles | 第一原理
The derivative of f(x) from first principles is defined by the limit of the difference quotient as h tends to zero:
函数 f(x) 从第一原理定义的导数是当 h 趋近于零时差商的极限：
f'(x) = lim(h→0) [f(x+h) − f(x)] / h
To use this definition, substitute f(x+h) and f(x), simplify, and then let h tend to 0. This is a common proof question in Edexcel Pure Mathematics.
使用这个定义时，先代入 f(x+h) 和 f(x)，化简，然后让 h 趋于 0。这是 Edexcel 纯数学中常见的证明题。
Example: for f(x) = x²,
例如：对于 f(x) = x²，
f'(x) = lim(h→0) [(x+h)² − x²] / h = lim(h→0) (2xh + h²) / h = lim(h→0) (2x + h) = 2x
This confirms that the derivative of x² is 2x.
这验证了 x² 的导数是 2x。
3. Power Rule | 幂法则
For any real number n, the derivative of xⁿ is n xⁿ⁻¹. This is the most frequently used rule in A-Level differentiation.
对于任意实数 n，xⁿ 的导数是 n xⁿ⁻¹。这是 A-Level 微分中最常用的法则。
d/dx (xⁿ) = n xⁿ⁻¹
Examples: d/dx (x⁵) = 5x⁴; d/dx (x⁻²) = −2x⁻³; d/dx (√x) = d/dx (x¹/²) = ½ x⁻¹/² = 1/(2√x).
例如：d/dx (x⁵) = 5x⁴；d/dx (x⁻²) = −2x⁻³；d/dx (√x) = d/dx (x¹/²) = ½ x⁻¹/² = 1/(2√x)。
When the power is negative or fractional, the same rule applies. This is essential for differentiating expressions with roots or denominators.
当指数为负数或分数时，同样的法则仍然适用。这对于求含根式或分母的表达式的导数至关重要。
4. Sum, Difference and Constant Multiple Rules | 和差与常数倍法则
Differentiation is linear, so you can differentiate term by term. If f(x) and g(x) are functions and k is a constant, then:
微分是线性的，因此你可以逐项求导。如果 f(x) 和 g(x) 是函数，k 是常数，则：
d/dx [f(x) ± g(x)] = f'(x) ± g'(x) and d/dx [k f(x)] = k f'(x)
Example: differentiate y = 5x⁴ − 3x² + 7.
例如：对 y = 5x⁴ − 3x² + 7 求导。
dy/dx = 20x³ − 6x + 0 = 20x³ − 6x
The derivative of a constant is always zero, because a constant function has zero rate of change.
常数的导数始终为零，因为常数函数的变化率为零。
5. Differentiating Common Functions | 常见函数的导数
You need to memorise the derivatives of the standard trigonometric, exponential and logarithmic functions. These appear frequently in Edexcel questions.
你需要熟记标准三角函数、指数函数和对数函数的导数。这些在 Edexcel 考题中频繁出现。
| f(x) | f'(x) | 中文说明 |
|---|---|---|
| sin x | cos x | 正弦的导数是余弦 |
| cos x | −sin x | 余弦的导数是负正弦 |
| tan x | sec²x | 正切的导数是正割平方 |
| eˣ | eˣ | eˣ 的导数仍是 eˣ |
| ln x | 1/x | ln x 的导数是 1/x |
For example, d/dx (sin x + eˣ) = cos x + eˣ. You should be able to quote these results without deriving them in the exam.
例如，d/dx (sin x + eˣ) = cos x + eˣ。在考试中你应该能够直接引用这些结果，无需重新推导。
6. Chain Rule | 链式法则
The chain rule is used when differentiating a composite function, that is, a function of a function. If y = g(h(x)), then dy/dx = g'(h(x)) × h'(x). In Leibniz notation, dy/dx = dy/du × du/dx.
链式法则用于求复合函数（即函数的函数）的导数。如果 y = g(h(x))，则 dy/dx = g'(h(x)) × h'(x)。用莱布尼茨记号写为 dy/dx = dy/du × du/dx。
For powers of a function: if y = [f(x)]ⁿ, then dy/dx = n [f(x)]ⁿ⁻¹ f'(x). This is the power rule extended by the chain rule.
对于函数的幂：如果 y = [f(x)]ⁿ，则 dy/dx = n [f(x)]ⁿ⁻¹ f'(x)。这是幂法则经链式法则扩展后的形式。
Example 1: y = (3x² + 5)⁴.
例 1：y = (3x² + 5)⁴。
dy/dx = 4(3x² + 5)³ × 6x = 24x(3x² + 5)³
Example 2: y = sin(2x).
例 2：y = sin(2x)。
dy/dx = 2 cos(2x)
Always remember to multiply by the derivative of the inner function. Forgetting this factor is one of the most common errors in differentiation.
始终记住要乘以内层函数的导数。忘记这个因子是微分中最常见的错误之一。
7. Product Rule | 乘积法则
If y = u(x) v(x), where u and v are both functions of x, then the derivative is given by:
如果 y = u(x) v(x)，其中 u 和 v 都是 x 的函数，则导数为：
dy/dx = u'(x) v(x) + u(x) v'(x)
In words: differentiate the first function, leave the second alone, plus leave the first alone, differentiate the second.
用语言描述：先对第一个函数求导，第二个保持不变；再加上第一个保持不变，对第二个函数求导。
Example: y = x² sin x. Let u = x² and v = sin x, then u’ = 2x and v’ = cos x.
例：y = x² sin x。令 u = x²，v = sin x，则 u’ = 2x，v’ = cos x。
dy/dx = 2x sin x + x² cos x
Do not simply multiply the two derivatives together; the product rule is not (uv)’ = u’v’.
不要简单地将两个导数相乘；(uv)’ 不等于 u’v’。
8. Quotient Rule | 商法则
If y = u(x) / v(x), where v(x) ≠ 0, then the derivative is:
如果 y = u(x) / v(x)，其中 v(x) ≠ 0，则导数为：
dy/dx = [u'(x) v(x) − u(x) v'(x)] / [v(x)]²
An easy way to remember the numerator is ‘v du minus u dv’, all over v squared.
记忆分子的简单方法是 ‘v du 减 u dv’，然后再除以 v 的平方。
Example: y = x / (x² + 1). Let u = x, v = x² + 1, so u’ = 1 and v’ = 2x.
例：y = x / (x² + 1)。令 u = x，v = x² + 1，则 u’ = 1，v’ = 2x。
dy/dx = [1(x² + 1) − x(2x)] / (x² + 1)² = (1 − x²) / (x² + 1)²
Be careful with signs in the numerator. A common mistake is to write u v’ − u’ v instead of u’ v − u v’.
注意分子中的符号。常见错误是把 u v’ − u’ v 写成 u’ v − u v’ 的顺序颠倒。
9. Derivatives of Exponential and Logarithmic Functions | 指数与对数函数的导数
The exponential function eˣ has the special property that its derivative is itself. More generally, for a constant k:
指数函数 eˣ 具有一个特殊性质：它的导数就是它本身。更一般地，对于常数 k：
d/dx (eᵏˣ) = k eᵏˣ
For a general exponential base a, d/dx (aˣ) = aˣ ln a. The natural logarithm function has derivative 1/x for x > 0, and by the chain rule, d/dx [ln(kx)] = 1/x.
对于一般的指数底数 a，d/dx (aˣ) = aˣ ln a。自然对数函数的导数是 1/x（x > 0），根据链式法则，d/dx [ln(kx)] = 1/x。
Example: y = e³ˣ + ln(2x).
例：y = e³ˣ + ln(2x)。
dy/dx = 3e³ˣ + 1/x
Remember that ln(kx) can also be written as ln k + ln x, and the derivative of ln k is zero, so the result is 1/x.
记住 ln(kx) 也可以写成 ln k + ln x，而 ln k 的导数为零，因此结果就是 1/x。
10. Second Derivatives and Higher Derivatives | 二阶及高阶导数
The second derivative of f(x), written f”(x) or d²y/dx², is the derivative of f'(x). It measures the rate of change of the gradient, which is useful for determining concavity and classifying stationary points.
函数 f(x) 的二阶导数写作 f”(x) 或 d²y/dx²，它是 f'(x) 的导数。它衡量斜率的变化率，常用于判断凹凸性和对驻点进行分类。
Example: f(x) = 4x³ − 6x² + 2x − 7.
例：f(x) = 4x³ − 6x² + 2x − 7。
f'(x) = 12x² − 12x + 2 and f”(x) = 24x − 12
You may also be asked for the third derivative f”'(x) or higher derivatives. Edexcel questions sometimes require finding f”(x) after applying
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