📚 Differentiating Functions with Two or More Terms | 含两项或多项函数的微分
In A-Level Mathematics, many functions you encounter are built from several simpler parts added or subtracted together. To differentiate such functions efficiently, you learn to apply basic rules to each term individually. This article consolidates the key skills needed for term-by-term differentiation, covering polynomials, negative and fractional powers, trigonometric and exponential terms, and the practical steps of simplifying expressions before you differentiate. Understanding these foundations will prepare you for product, quotient, and chain rules later.
在 A-Level 数学中，你遇到的许多函数都是由若干个简单部分相加或相减组合而成的。要高效地对这些函数进行微分，你需要学会对每一项分别应用基本法则。本文将归纳逐项微分所需的关键技巧，涵盖多项式、负指数和分数指数、三角函数项与指数函数项，以及在微分前先化简表达式的实用步骤。掌握这些基础将为后续学习积法则、商法则和链式法则奠定扎实的基础。
1. Review of Basic Differentiation: The Power Rule | 基础微分回顾：幂法则
The power rule is the fundamental tool for differentiating terms of the form axⁿ. If f(x) = xⁿ where n is any real number, then f'(x) = n xⁿ⁻¹. When a constant coefficient a multiplies the power, the derivative becomes a n xⁿ⁻¹. This rule can be repeatedly applied to any term where the variable is raised to a power.
幂运算是微分形如 axⁿ 的项的基本工具。若 f(x) = xⁿ，其中 n 为任意实数，则 f'(x) = n xⁿ⁻¹。当存在常数系数 a 时，导数为 a n xⁿ⁻¹。这一法则可反复应用于任意变量带幂次的项。
d/dx (xn) = n xn-1
Example: Differentiate 7x⁴ own the power 4 to multiply the coefficient: 7 × 4 = 28, then reduce the power by 1: 28x³.
例子：对 7x⁴ 求导。将指数 4 下乘到系数：7 × 4 = 28，再把指数减 1 得 28x³。
Example with constant term: The derivative of any constant is zero, because a constant can be thought of as c x⁰, and applying the power rule gives c × 0 × x⁻¹ = 0.
常数项例子：任意常数的导数为零，因为常数可看作 c x⁰，应用幂法则得 c × 0 × x⁻¹ = 0。
2. The Sum and Difference Rule: Differentiating Term by Term | 和差法则：逐项微分
When a function is expressed as a sum or difference of several simpler functions, you can differentiate each term separately and then combine the results. Mathematically, if f(x) = u(x) ± v(x), then f'(x) = u'(x) ± v'(x). This rule allows you to break down complicated expressions into manageable pieces.
当一个函数表示为若干个简单函数之和或差时，你可以分别对每一项进行微分，再将结果合并。从数学上讲，若 f(x) = u(x) ± v(x)，则 f'(x) = u'(x) ± v'(x)。这一法则让你能把复杂表达式拆分为容易处理的小块。
For example, given f(x) = 5x³ − 2x² + 4x − 9, apply the power rule to each term:
例如，给定 f(x) = 5x³ − 2x² + 4x − 9，依次对每一项应用幂法则：
a) d/dx (5x³) = 15x²
b) d/dx (−2x²) = −4x
c) d/dx (4x) = 4
d) d/dx (−9) = 0
a) d/dx (5x³) = 15x²
b) d/dx (−2x²) = −4x
c) d/dx (4x) = 4
d) d/dx (−9) = 0
Thus, f'(x) = 15x² − 4x + 4.
因此，f'(x) = 15x² − 4x + 4。
3. Step-by-Step Differentiation of Polynomials | 多项式微分步骤详解
Polynomials consist entirely of terms of the form a xⁿ where n is a positive integer or zero. The procedure is systematic: identify each term, differentiate it individually, and then write the sum of the derivatives while keeping the same signs. It is helpful to rewrite the polynomial in descending powers of x to avoid missing any term.
多项式完全由形如 a xⁿ（n 为正整数或零）的项组成。其操作步骤很系统化：找出每一项，逐项微分，然后将导数求和并保持原有的正负号。建议先将多项式按 x 的降幂排列，以免遗漏某项。
Step 1: Write the polynomial with all terms clearly visible, e.g., f(x) = 2x⁵ + 0x⁴ − 3x³ + 7x − 12. (The zero term is optional but can be used for alignment.)
第一步：写出多项式的每一项并确保清晰可见，例如 f(x) = 2x⁵ + 0x⁴ − 3x³ + 7x − 12。（零次项可选，但有助于对齐。）
Step 2: Differentiate the highest power first: d/dx (2x⁵) = 10x⁴.
第二步：从最高次项开始微分：d/dx (2x⁵) = 10x⁴.
Step 3: Move to the next term, −3x³, giving −9x².
第三步：接着微分 −3x³，得 −9x²。
Step 4: The linear term 7x becomes 7.
第四步：线性项 7x 变为 7。
Step 5: The constant −12 becomes 0.
第五步：常数 −12 变为 0。
Step 6: Assemble the derivative: f'(x) = 10x⁴ − 9x² + 7.
第六步：组合成导数：f'(x) = 10x⁴ − 9x² + 7。
4. Dealing with Negative and Fractional Powers | 处理负指数与分数指数
ve or fractional exponents so that the power rule still applies. For instance, 1/x² = x⁻², √x = x1/2, and 1/√x = x−1/2. Once in this form, you can differentiate exactly as you would for a polynomial term.
许多包含根号或分母中含有变量的函数，可以通过改写为负指数或分数指数形式，使幂法则仍然适用。例如，1/x² = x⁻²，√x = x1/2，以及 1/√x = x−1/2。转化为这种形式后，就可以像多项式项一样进行微分。
Example: Differentiate f(x) = 4/x³ − 3√x + 5x⁻². First, express everything with powers of x: f(x) = 4x⁻³ − 3x1/2 + 5x⁻².
举例：对 f(x) = 4/x³ − 3√x + 5x⁻² 求导。首先将所有项写成 x 的幂形式：f(x) = 4x⁻³ − 3x1/2 + 5x⁻²。
Differentiate term by term:
逐项微分：
d/dx (4x⁻³) = 4(−3)x⁻⁴ = −12x⁻⁴
d/dx (4x⁻³) = 4(−3)x⁻⁴ = −12x⁻⁴
d/dx (−3x1/2) = −3 × (1/2) x−1/2 = −(3/2)x−1/2
d/dx (−3x1/2) = −3 × (1/2) x−1/2 = −(3/2)x−1/2
d/dx (5x⁻²) = 5(−2)x⁻³ = −10x⁻³
d/dx (5x⁻²) = 5(−2)x⁻³ = −10x⁻³
Therefore, f'(x) = −12x⁻⁴ − (3/2)x−1/2 − 10x⁻³. You may convert back to rational form if required.
因此，f'(x) = −12x⁻⁴ − (3/2)x−1/2 − 10x⁻³。若需要，可再转回分式和根式表达。
5. Differentiating Sums Involving Trigonometry and Exponentials | 涉及三角函数和指数函数的和差微分
When a function contains sin x, cos x, tan x, or eˣ alongside algebraic terms, the sum rule still holds. You simply need to know the standard derivatives:
当函数中含有 sin x、cos x、tan x 或 eˣ 以及代数项时，和差法则依然成立。你只需熟记这些标准导数：
| d/dx (sin x) = cos x | sin x 的导数是 cos x |
| d/dx (cos x) = −sin x | cos x 的导数是 −sin x |
| d/dx (tan x) = sec² x | tan x 的导数是 sec² x |
| d/dx (eˣ) = eˣ | eˣ 的导数是 eˣ |
| d/dx (ln x) = 1/x | ln x 的导数是 1/x |
Example: f(x) = 2 sin x + 3eˣ − 4x². Differentiate each term: derivative of 2 sin x is 2 cos x; of 3eˣ is 3eˣ; of −4x² is −8x. So f'(x) = 2 cos x + 3eˣ − 8x.
例子：f(x) = 2 sin x + 3eˣ − 4x²。逐项求导：2 sin x 的导数为 2 cos x；3eˣ 的导数为 3eˣ；−4x² 的导数为 −8x。故 f'(x) = 2 cos x + 3eˣ − 8x。
Note: If the argument of a trigonometric or exponential function is a multiple of x alone (e.g., sin 2x), you will later use the chain rule, but the sum rule remains the same for combining distinct terms.
注意：如果三角函数或指数函数的自变量是 x 的倍数（如 sin 2x），后续会用到链式法则，但合并不同项时和差法则依然不变。
6. Expand Before You Differentiate | 先展开再求导
If you are given a product of two or more brackets, such as (x+2)(x−3) or (x²+1)(x−4), it is usually quicker to expand the expression into a polynomial of separate terms before differentiating. This avoids having to use the product rule unnecessarily and often reduces the risk of algebraic mistakes.
如果给出的函数是两个或多个括号的乘积，例如 (x+2)(x−3) 或 (x²+1)(x−4)，通常更快的做法是先将表达式展开为一个由若干单独项组成的多项式，再进行微分。这样可以避免不必要地使用积法则，而且往往能减少代数操作失误的风险
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