SQA Advanced Higher Maths Formula & Theorem Quick Reference | SQA 高阶数学公式定理速查手册

📚 SQA Advanced Higher Maths Formula & Theorem Quick Reference | SQA 高阶数学公式定理速查手册

This quick reference handbook provides essential formulae, theorems and techniques required for the SQA Advanced Higher Mathematics course. It covers differentiation, integration, complex numbers, vectors, matrices, sequences, differential equations, proof by induction and more. Each section is presented with a concise English explanation immediately followed by its Chinese equivalent to support bilingual learning and rapid revision.

本速查手册汇集了 SQA 高阶数学（Advanced Higher Mathematics）课程的核心公式、定理与方法，涵盖微分、积分、复数、向量、矩阵、数列、微分方程、归纳法证明等重点知识。每个要点均先给出英文说明，随后配以中文翻译，方便双语对照学习与快速复习。

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1. Differentiation Rules | 微分法则

The derivative of a function describes its instantaneous rate of change. Mastery of the basic rules is essential for tackling more complex problems.

函数的导数描述其瞬时变化率。掌握基本求导法则是解决复杂问题的基础。

d/dx (xⁿ) = n xⁿ⁻¹ (n ≠ 0)

The power rule is the foundation of polynomial differentiation. For f(x) = xⁿ, the derivative is n·xⁿ⁻¹.

幂函数求导法则是多项式微分的基础。若 f(x) = xⁿ，则导数为 n·xⁿ⁻¹。

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

Product Rule: The derivative of a product of two functions is the first function times the derivative of the second plus the second times the derivative of the first.

乘积法则：两个函数乘积的导数等于第一个函数乘第二个函数的导数，加上第二个函数乘第一个函数的导数。

d/dx (u/v) = (v du/dx − u dv/dx) / v²

Quotient Rule: The derivative of a quotient follows ‘low d-high minus high d-low over low-squared’.

商法则：分子乘分母的导数减去分母乘分子的导数，整体除以分母的平方。

dy/dx = dy/du · du/dx

Chain Rule: Used when differentiating composite functions. Identify an inner function u and multiply the derivatives.

链式法则：用于复合函数的求导。选定内层函数 u，将导数相乘。

d/dx (sin x) = cos x,   d/dx (cos x) = −sin x

d/dx (tan x) = sec² x,   d/dx (cot x) = −csc² x

Derivatives of basic trigonometric functions. The derivative of tan x is sec² x, where sec x = 1/cos x.

基本三角函数的导数。tan x 的导数为 sec² x，其中 sec x = 1/cos x。

d/dx (eˣ) = eˣ,   d/dx (aˣ) = aˣ ln a

d/dx (ln x) = 1/x

Exponential and logarithmic derivatives. The natural exponential function is its own derivative; log base a requires the natural logarithm factor.

指数函数与对数函数的导数。自然指数函数的导数等于其本身；以 a 为底的指数函数需乘以 ln a。

d/dx (sin⁻¹ x) = 1 / √(1−x²)

d/dx (cos⁻¹ x) = −1 / √(1−x²)

d/dx (tan⁻¹ x) = 1 / (1 + x²)

Derivatives of the inverse trigonometric functions. These are essential for integration as well.

反三角函数的导数。这些公式在积分中同样至关重要。

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2. Integration Techniques | 积分方法

Integration reverses differentiation. Here we summarise standard integrals, key techniques, and useful substitutions.

积分是微分的逆运算。以下汇总标准积分、核心积分技巧与常用代换方法。

∫ xⁿ dx = xⁿ⁺¹ / (n+1) + C   (n ≠ −1)

∫ 1/x dx = ln |x| + C

The power rule for integration, with the special case of the reciprocal function giving a natural logarithm.

幂函数的积分法则，倒数函数的积分则为自然对数。

∫ eˣ dx = eˣ + C,   ∫ aˣ dx = aˣ / ln a + C

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