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Edexcel A-Level Pure Maths: Differentiation Rules and Applications | 爱德思A-Level纯数学：微分法则与应用

2026-08-29 17:55 · 本文部分内容由AI辅助创作

📚 Edexcel A-Level Pure Maths: Differentiation Rules and Applications | 爱德思A-Level纯数学：微分法则与应用

Differentiation is one of the most heavily examined topics in the Edexcel A-Level Pure Mathematics specification. It develops the idea of a gradient from straight lines to curves, and it provides the mathematical machinery for optimisation, curve sketching, kinematics and many applied problems.

微分是爱德思A-Level纯数学大纲中考查最频繁的主题之一。它将直线的梯度概念扩展到曲线，并为最优化、曲线作图、运动学以及许多应用问题提供了数学工具。

1. The Limit Definition | 极限定义

Differentiation from first principles defines the derivative of a function f at a point x as the limit of the gradient of a chord joining (x, f(x)) and (x+h, f(x+h)) as h tends to zero. This formal definition is important because it links the derivative to the idea of instantaneous rate of change.

从第一性原理求导将函数 f 在点 x 处的导数定义为连接 (x, f(x)) 与 (x+h, f(x+h)) 的割线斜率在 h 趋于零时的极限。这个正式定义之所以重要，是因为它将导数与瞬时变化率的概念联系起来。

f'(x) = lim h→0 [f(x+h) − f(x)] / h

on gives f(x+h) − f(x) = 2xh + h². Dividing by h and letting h → 0 leaves the derivative f'(x) = 2x.

例如，若 f(x) = x²，展开后得到 f(x+h) − f(x) = 2xh + h²。除以 h 并令 h → 0，就得到导数 f'(x) = 2x。

2. Power Rule and Sum Rule | 幂函数法则与加减法则

The power rule states that if f(x) = xⁿ, then f' is the foundation for differentiating polynomials and many rational or radical functions.

幂函数法则指出，若 f(x) = xⁿ，则 f'(x) = n xⁿ⁻¹，其中 n 是任意实数常数。这个法则是求多项式以及许多有理函数或根式函数导数的基础。

d/dx (xⁿ) = n xⁿ⁻¹

rivative of a sum is the sum of the derivatives. Therefore d/dx [5x³ − 2x + 7] = 15x² − 2.

常数与和式可以分别处理：常数的导数为零，和式的导数等于各项导数的和。因此 d/dx [5x³ − 2x + 7] = 15x² − 2。

3. Chain Rule | 链式法则

osed with another. If y = f(u) and u = g(x), then dy/dx is the product of dy/du and du/dx.

链式法则用于一个函数与另一个函数复合的情况。若 y = f(u) 且 u = g(x)，则 dy/dx 等于 dy/du 与 du/dx 的乘积。

dy/dx = dy/du × du/dx

+ 1, so y = u⁵. Then dy/du = 5u⁴ and du/dx = 6x. Multiplying gives dy/dx = 30x(3x² + 1)⁴.

例如，设 y = (3x² + 1)⁵。令 u = 3x² + 1，则 y = u⁵。于是 dy/du = 5u⁴，du/dx = 6x。相乘得到 dy/dx = 30x(3x² + 1)⁴。

4. Product Rule | 乘法法则

ects you to quote and apply the product rule accurately. The derivative is u times the derivative of v plus v times the derivative of u.

当对两个函数的乘积 y = u(x)v(x) 求导时，爱德思考试要求准确引用并应用乘法法则。导数等于 u 乘以 v 的导数再加上 v 乘以 u 的导数。

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

e u = x² and v = sin x. Then du/dx = 2x and dv/dx = cos x, so dy/dx = 2x sin x + x² cos x.

对于 y = x² sin x，取 u = x²，v = sin x。则 du/dx = 2x，dv/dx = cos x，所以 dy/dx = 2x sin x + x² cos x。

5. Quotient Rule | 除法法则

is found using the denominator squared in the denominator of the result. The numerator is v du/dx minus u dv/dx.

对于商式 y = u/v，其导数的分母为原分母的平方，分子为 v du/dx 减去 u dv/dx。

d/dx (u/v) = (v du/dx − u dv/dx) / v²

rivative is (x² + 1) − x(2x) all over (x² + 1)², which simplifies to (1 − x²) / (x² + 1)².

若 y = x / (x² + 1)，则 u = x，v = x² + 1，du/dx = 1，dv/dx = 2x。导数为 [(x² + 1) − x(2x)] / (x² + 1)²，化简为 (1 − x²) / (x² + 1)²。

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6. Exponentials and Logarithms | 指数函数与对数函数求导

Exponential and logarithmic functions have distinctive derivatives. The exponential function eˣ is its own derivative, while the natural logarithm ln x differentiates to the reciprocal of x.

指数函数与对数函数有独特的导数。指数函数 eˣ 的导数就是它本身，而自然对数 ln x 的导数为 x 的倒数。

d/dx (eˣ) = eˣ
d/dx (ln x) = 1/x

For a composite exponential such as y = eᵏˣ, the chain rule gives dy/dx = k eᵏˣ. Similarly, d/dx [ln(kx)] = 1/x because the constant k cancels after differentiation.

对于像 y = eᵏˣ 这样的复合指数函数，链式法则给出 dy/dx = k eᵏˣ。类似地，d/dx [ln(kx)] = 1/x，因为常数 k 在求导后会被消去。

Function f(x)Derivative f'(x)
eˣeˣ
eᵏˣk eᵏˣ
ln x1/x
aˣaˣ ln a

7. Trigonometric Differentiation | 三角函数求导

The standard trigonometric derivatives must be memorised accurately. Edexcel often tests these results in combination with the product, quotient and chain rules.

标准的三角函数导数必须准确记忆。爱德思考试经常将这些结果与乘法法则、除法法则和链式法则结合起来考查。

d/dx (sin x) = cos x
d/dx (cos x) = −sin x
d/dx (tan x) = sec² x

= (sin x)². Using the chain rule gives dy/dx = 2 sin x cos x, which is the same as sin 2x.

例如，y = sin² x 可以看作 y = (sin x)²。使用链式法则得到 dy/dx = 2 sin x cos x，这也等于 sin 2x。

8. Implicit Differentiation | 隐函数求导

When a relationship between x and y is given implicitly, differentiate both sides with respect to x, treating y as a function of x. Every derivative of a y-term must include a factor dy/dx.

当 x 与 y 的关系以隐函数形式给出时，对等式两边关于 x 求导，并将 y 视为 x 的函数。每个含 y 的项求导后都必须乘上因子 dy/dx。

d/dx (y²) = 2y dy/dx

y² = 25, differentiating both sides gives 2x + 2y dy/dx = 0. Solving gives dy/dx = −x / y.

对于圆 x² + y² = 25，对两边求导得到 2x + 2y dy/dx = 0。解得 dy/dx = −x / y。