Mastering Differentiation for Edexcel A-Level Maths | 掌握 Edexcel A-Level 数学微分法

📚 Mastering Differentiation for Edexcel A-Level Maths | 掌握 Edexcel A-Level 数学微分法

Differentiation is a central topic in the Edexcel A-Level Mathematics specification. It describes how a function changes at any point and is essential for understanding rates of change, curve sketching and optimisation problems.

微分是 Edexcel A-Level 数学大纲中的核心主题。它描述函数在任意点的变化情况，对于理解变化率、曲线作图以及最优化问题都至关重要。

1. The Idea of a Derivative | 导数的概念

The derivative of a function f(x) at a point x = a gives the gradient of the tangent to the curve y = f(x) at that point. It is also the instantaneous rate of change of y with respect to x.

函数 f(x) 在 x = a 处的导数给出了曲线 y = f(x) 在该点切线的斜率。它也表示 y 关于 x 的瞬时变化率。

If the derivative is positive, the function is increasing; if it is negative, the function is decreasing. This sign interpretation is often tested in exam questions.

如果导数为正，函数在该点递增；如果导数为负，函数在该点递减。这种符号含义经常在考试题中出现。

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:

导数的正式定义是差商的极限：

f'(x) = lim h → 0 [f(x + h) − f(x)] / h

In Edexcel exams, you may be asked to use this definition for simple functions such as f(x) = x² or f(x) = x³. Start by writing f(x + h), subtract f(x), simplify the fraction, and then let h tend to zero.

在 Edexcel 考试中，你可能会被要求对简单函数（如 f(x) = x² 或 f(x) = x³）使用该定义。首先写出 f(x + h)，减去 f(x)，化简分式，然后令 h 趋于零。

3. Notation for Derivatives | 导数记号

Edexcel uses several notations: f'(x), dy/dx, and d/dx [f(x)]. All mean the same thing but may appear in different contexts, especially when y is given explicitly.

Edexcel 会使用几种记号：f'(x)、dy/dx 以及 d/dx [f(x)]。它们的含义相同，但在不同语境中出现，尤其是当 y 被明确给出时。

The second derivative, f”(x) or d²y/dx², measures the rate of change of the gradient and helps determine concavity and the nature of stationary points.

二阶导数 f”(x) 或 d²y/dx² 衡量斜率的变化率，有助于判断凹凸性和驻点的性质。

4. Power Rule | 幂函数求导法则

For f(x) = xⁿ, the derivative is f'(x) = nxⁿ⁻¹. This is the most frequently used rule in A-Level differentiation.

对于 f(x) = xⁿ，其导数为 f'(x) = nxⁿ⁻¹。这是 A-Level 微分中最常用的法则。

Remember to rewrite expressions such as 1/x, √x and 1/x² as x⁻¹, x¹ᐟ² and x⁻² before applying the rule.

请记住先将 1/x、√x 和 1/x² 等表达式改写为 x⁻¹、x¹ᐟ² 和 x⁻²，然后再应用该法则。

5. Constant Multiple and Sum Rules | 常数倍与和差法则

If f(x) = ag(x), where a is a constant, then f'(x) = ag'(x). Also, the derivative of a sum or difference is the sum or difference of the derivatives.

如果 f(x) = ag(x)（其中 a 为常数），则 f'(x) = ag'(x)。同样，和或差的导数等于导数的和或差。

For example, d/dx (3x⁴ − 5x² + 2x − 7) = 12x³ − 10x + 2. Constants disappear because their rate of change is zero.

例如，d/dx (3x⁴ − 5x² + 2x − 7) = 12x³ − 10x + 2。常数项消失是因为它的变化率为零。

6. Product Rule | 乘积法则

For y = uv, where u and v are functions of x, the derivative is:

对于 y = uv，其中 u 和 v 都是 x 的函数，导数为：

dy/dx = u dv/dx + v du/dx

It is helpful to write u, v, du/dx and dv/dx separately before substituting into the formula. This reduces sign errors and improves clarity.

建议先分别写出 u、v、du/dx 和 dv/dx，再代入公式。这样可以减少符号错误并提高清晰度。

7. Quotient Rule | 商法则

For y = u/v, the quotient rule states:

对于 y = u/v，商法则为：

dy/dx = (v du/dx − u dv/dx) / v²

Pay close attention to the order of terms in the numerator: v du/dx comes first, then minus u dv/dx. Many exam errors come from reversing this order.

请特别注意分子中各项的顺序：先写 v du/dx，再减去 u dv/dx。许多考试错误都源于顺序颠倒。

8. Chain Rule | 链式法则

The chain rule is used for composite functions y = f(g(x)). It states:

链式法则用于复合函数 y = f(g(x))。其内容为：

dy/dx = dy/du × du/dx

Common examples include y = (3x + 2)⁵, y = e²ˣ and y = sin(5x). Identify the inner function u, differentiate it, then multiply by the derivative of the outer function with respect to u.

常见例子包括 y = (3x + 2)⁵、y = e²ˣ 和 y = sin(5x)。确定内层函数 u，对其求导，然后乘以外层函数关于 u 的导数。

9. Derivatives of Trigonometric Functions | 三角函数导数

The standard results are d/dx (sin x) = cos x, d/dx (cos x) = −sin x, and d/dx (tan x) = sec² x. These results only apply when x is measured in radians; convert degrees to radians first.

标准结果是 d/dx (sin x) = cos x、d/dx (cos x) = −sin x 以及 d/dx (tan x) = sec² x。这些结果仅在 x 以弧度为单位时成立；如果角度以度为单位，请先转换为弧度。

For functions like y = sin(ax + b), combine the trigonometric result with the chain rule to obtain dy/dx = a cos(ax + b).

对于 y = sin(ax + b) 这类函数，将三角函数求导结果与链式法则结合，可得 dy/dx = a cos(ax + b)。

10. Derivatives of Exponential and Logarithmic Functions | 指数与对数函数导数

The derivative of eˣ is eˣ. More generally, d/dx (eᵏˣ) = keᵏˣ, where k is a constant.

eˣ 的导数是 eˣ。更一般地，d/dx (eᵏˣ) = keᵏˣ，其中 k 为常数。

For natural logarithms, d/dx (ln x) = 1/x for x > 0. For y = ln(kx), the derivative is still 1/x after applying the chain rule, because the constant factor cancels.

对于自然对数，当 x > 0 时 d/dx (ln x) = 1/x。对于 y = ln(kx)，应用链式法则后导数仍为 1/x，因为常数因子会抵消。

11. Stationary Points and Graphical Interpretation | 驻点与图像解释

Stationary points occur where f'(x) = 0. To determine whether a stationary point is a local maximum, local minimum or point of inflection, use the second derivative or a sign table for f'(x).

驻点出现在 f'(x) = 0 的位置。要判断驻点是局部极大值、局部极小值还是拐点，可以使用二阶导数或 f'(x) 的符号表。

If f”(x) < 0, the stationary point

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Mastering Differentiation for Edexcel A-Level Maths | 掌握 Edexcel A-Level 数学微分法