Differentiation: Rules, Techniques and Applications | 微分：法则、技巧与应用

📚 Differentiation: Rules, Techniques and Applications | 微分：法则、技巧与应用

In Edexcel A-Level Mathematics, differentiation is a central Pure topic that connects gradients, rates of change, curve sketching and optimisation. It appears in both AS and A2 papers, and it is also the tool that later topics such as integration, numerical methods and mechanics depend on. This revision guide explains the key rules, the standard derivatives you must memorise, the techniques examined at A2, and the exam habits that convert method marks into full marks.

在 Edexcel A-Level 数学中，微分是纯数学的核心主题，连接斜率、变化率、曲线作图与最优化。它既出现在 AS 试卷，也贯穿 A2 试卷，并且是后续积分、数值方法与力学等主题的基础工具。本复习指南讲解关键法则、必须熟记的标准导数、A2 阶段考查的技巧，以及能把方法分转化为满分的考试习惯。

---

1. The Idea of Differentiation | 微分的基本思想

res the instantaneous rate of change of a function. For a curve y = f(x), the derivative f' a chord joining two nearby points on the curve, taken as the horizontal gap shrinks towards zero.

微分衡量函数在某一点的瞬时变化率。对于曲线 y = f(x)，导数 f'(x)，也写作 dy/dx，表示曲线在横坐标为 x 处切线的斜率。从几何上看它是一个极限：曲线上两点连成的弦的斜率，在水平间距趋于 0 时的极限。

The average rate of change over an interval of width h is [f(x + h) − f(x)] / h. Letting h approach zero turns that average into the instantaneous rate of change, which is the formal definition of the derivative.

区间宽度为 h 时的平均变化率为 [f(x + h) − f(x)] / h。令 h 趋于 0，平均变化率就转化为瞬时变化率，这正是导数的形式化定义。

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

Several notations mean the same thing and examiners accept all of them: f'(x) and dy/dx are the most common, while d²y/dx² denotes the second derivative. A question phrased as "find the rate of change of y with respect to x" or "find the gradient function" is simply asking for the derivative.

有多种记号含义相同，考官都接受：最常用的是 f'(x) 与 dy/dx，而 d²y/dx² 表示二阶导数。若题目表述为 "求 y 关于 x 的变化率" 或 "求斜率函数"，实际上就是要求导数。

A useful physical reading: if s is displacement and t is time, then ds/dt is velocity and d²s/dt² is acceleration. This is why differentiation appears so often in mechanics questions.

一个有用的物理理解：若 s 表示位移、t 表示时间，则 ds/dt 是速度，d²s/dt² 是加速度。这正是微分频繁出现在力学题中的原因。

---

2. First Principles and the Power Rule | 第一原理与幂函数法则

h to get 2x + h, then let h → 0 to obtain 2x. The same three steps (expand, cancel h, take the limit) work for any polynomial, and the mark scheme wants all three written down.

Edexcel 考试可能要求从第一原理出发对简单函数求导。以 f(x) = x² 为例，展开 f(x + h) − f(x) = (x + h)² − x² = 2xh + h²，除以 h 得到 2x + h，再令 h → 0，得到 2x。同样的三步（展开、约去 h、取极限）适用于任何多项式，评分标准要求三步都写出来。

Once the definition is understood, we normally apply the power rule, which is far faster and is the workhorse of the whole topic.

在理解了定义之后，通常直接使用幂函数法则，它快得多，也是整个主题的核心工具。

If y = xⁿ, then dy/dx = n xⁿ⁻¹

Multiplying by a constant simply multiplies the derivative, so the derivative of k xⁿ is k n xⁿ⁻¹. A constant term on its own differentiates to zero, because a horizontal line has gradient zero.

乘以常数只需把常数乘在导数上，因此 k xⁿ 的导数为 k n xⁿ⁻¹。单独的常数项求导为 0，因为水平直线的斜率为 0。

Differentiation is linear, which means you can differentiate a sum or difference term by term:

微分是线性的，这意味着可以逐项对和或差求导：

d/dx [ a f(x) ± b g(x) ] = a f'(x) ± b g'(x)

write the expression first. For example, y = 1/x = x⁻¹ gives dy/dx = −x⁻² = −1/x², and y = √x = x^½ gives dy/dx = ½ x^−½ = 1/(2√x). Turning roots and reciprocals into powers before differentiating is a habit that prevents most errors in this section.

只要先改写表达式，幂函数法则对负指数与分数指数同样成立。例如 y = 1/x = x⁻¹ 给出 dy/dx = −x⁻² = −1/x²；y = √x = x^½ 给出 dy/dx = ½ x^−½ = 1/(2√x)。先把根式和倒数化为幂再求导，可以避免本节大多数错误。

---

3. Standard Derivatives for Exponentials, Logarithms and Trigonometry | 指数、对数与三角函数的导数

You must memorise the standard derivatives below; they are not given in the formula booklet. In calculus, trigonometric functions are always in radians unless the question states otherwise, and your calculator must be in radian mode.

必须熟记以下标准导数，它们不会出现在公式册中。微积分中，若无特殊说明，三角函数一律使用弧度制，计算器也必须设置为弧度模式。

Function f(x)	Derivative f'(x)	Notes
eˣ	eˣ	slope equals height
ln x	1/x	x > 0 only
sin x	cos x	radians
cos x	−sin x	note the minus sign
tan x	sec² x	= 1/cos² x
aˣ	aˣ ln a	a > 0, a ≠ 1

For a linear inner function, use the chain rule shortcut, which removes the need to write out a substitution every time.

对于线性内层函数，可使用链式法则的快捷形式，无需每次写出换元过程。

d/dx [ sin(ax + b) ] = a cos(ax + b)   and   d/dx [ e^{ax + b} ] = a e^{ax + b}

Notice that the constant b never appears in the answer: only the coefficient a of x does.

因此 d/dx [ cos(3x) ] = −3 sin(3x)，d/dx [ tan(2x) ] = 2 sec²(2x)，d/dx [ ln(5x + 2) ] = 5/(5x + 2)。注意常数 b 不会出现在答案中，只有 x 的系数 a 会影响结果。

---

4. The Chain Rule | 链式法则

f(u); then the rate of change of y with respect to x is the product of the two linked rates.

链式法则处理复合函数，也就是"函数套函数"。设 u = g(x)，使 y = f(u)，则 y 关于 x 的变化率等于两个相关变化率之积。

dy/dx = dy/du × du/dx

广告 / Sponsored

For a power of a function, this takes a particularly useful form, sometimes called the "bring the power down, reduce it by one, multiply by the derivative of the inside" rule.

对于函数的幂，链式法则有一种特别实用的形式，可概括为"幂次前移、降一次、再乘内层函数的导数"。

d/dx [ f(x) ]ⁿ = n [ f(x) ]ⁿ⁻¹ × f'(x)

e outer power gives 5(3x² + 1)⁴, and the inner derivative is 6x, so dy/dx = 30x(3x² + 1)⁴.

例题：对 y = (3x² + 1)⁵ 求导。外层幂给出 5(3x² + 1)⁴，内层导数为 6x，因此 dy/dx = 30x(3x² + 1)⁴。

Other standard applications: d/dx [ e^x² ] = 2x e^x² ion question uses the chain rule at least once, often hidden inside a product or quotient.

其他常见应用：d/dx [ e^x² ] = 2x e^x²，d/dx [ ln(x² + 1) ] = 2x/(x² + 1)，d/dx [ sin(x³) ] = 3x² cos(x³)。A2 的微分题几乎都会用到链式法则，且常常隐藏在乘积或商之中。

Exam tip: before differentiating, say to yourself what the "outer" function is and what the "inner" and then substituting back is perfectly acceptable and often safer than doing it mentally.

考试技巧：求导前先在心里确认"外层"与"内层"分别是什么。写出 u = ... 再代回完全可接受，往往比心算更稳妥。

---

5. The Product Rule and Quotient Rule | 乘积法则与商法则

u v, where u and v both depend on x, then the derivative is the first times the derivative of the second, plus the second times the derivative of the first.

当两个函数相乘时使用乘积法则。若 y = u v，且 u、v 都依赖于 x，则导数为"前者乘后者的导数，加上后者乘前者的导数"。

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

ing the product rule in the same order every time is the simplest way to avoid sign and order slips.

例题：y = x² sin x。取 u = x²、v = sin x，则 du/dx = 2x、dv/dx = cos x，于是 dy/dx = 2x sin x + x² cos x。每次都用同一顺序书写乘积法则，是避免符号与顺序出错的最简单方法。

uotient rule is used for a fraction. If y = u/v where v is not zero, then the numerator is "bottom times derivative of top, minus top times derivative of bottom", all over the bottom squared.

商法则用于分数形式。若 y = u/v，且 v 不为 0，则分子为"分母乘分子的导数，减去分子乘分母的导数"，再除以分母的平方。

d/dx (u / v) = ( v du/dx − u dv/dx ) / v²

so du/dx = 1/x and dv/dx = 1. Hence dy/dx = [ x × (1/x) − ln x × 1 ] / x² = (1 − ln x)/x².

例题：y = (ln x)/x。此处 u = ln x、v = x，故 du/dx = 1/x、dv/dx = 1，因此 dy/dx = [ x × (1/x) − ln x × 1 ] / x² = (1 − ln x)/x²。

oduct rule is quicker. Second, the most common error is forgetting the minus sign in the quotient rule numerator, so double-check it under exam pressure.

两点实用提醒。第一，商总可以改写为负指数乘积，例如 (ln x)/x = x⁻¹ ln x，有时用乘积法则更快。第二，最常见的错误是忘记商法则分子中的负号，因此在考试压力下务必复核。

---

6. Tangents, Normals and Curve Sketching | 切线、法线与曲线作图

The gradient of the curve y = f(x) at x = a is simply f' e through (a, f(a)) with that gradient, so its equation follows from the point-slope form.

曲线 y = f(x) 在 x = a 处的斜率就是 f'(a)。该点的切线是过 (a, f(a)) 且斜率相同的直线，其方程可由点斜式写出。

Tangent: y − f(a) = f'(a) (x − a)

The normal is the line perpendicular to the tangent at the same point. Since perpendicular gradients multiply to −1, the gradient of the normal is −1 / f'(a), provided f'(a) is not zero.

法线是在同一点与切线垂直的直线。由于互相垂直的两条直线斜率之积为 −1，法线的斜率为 −1 / f'(a)，前提是 f'(a) 不为 0。

Normal: y − f(a) = ( −1 / f'(a) ) (x − a)

Worked example: y = x³ − 3x at the point where x = 2. Here f(2) = 8 − 6 = 2 and f'(x) = 3x² − 3, so f' − 2 = 9(x − 2), that is y = 9x − 16. The normal has gradient −1/9, giving y = −x/9 + 20/9.

例题：求 y = x³ − 3x 在 x = 2 处的切线与法线。f(2) = 8 − 6 = 2，f'(x) = 3x² − 3，故 f'(2) = 9。切线为 y − 2 = 9(x − 2)，即 y = 9x − 16；法线斜率为 −1/9，得 y = −x/9 + 20/9。

The sign of the derivative also tells you where a curve rises and falls: f'(x) > 0 means the function is increasing at that point, f'(x) < 0 means it is decreasing. Questions often ask you to find the set of values of x for which a function is increasing, which means solving the inequality f'(x) > 0, usually by factoring a quadratic and sketching its graph.

导数的符号还能说明曲线的升降：f'(x) > 0 表示函数在该点递增，f'(x) < 0 表示递减。常见题目要求求出使函数递增的 x 取值范围，即解不等式 f'(x) > 0，通常通过因式分解二次式并画出其图像来完成。

---

7. Stationary Points and the Second Derivative | 驻点与二阶导数

A stationary point is a point where the gradient is zero, that is f' he x-coordinates of the stationary points; substitute back into f(x) to get the y-coordinates, which many mark schemes require.

驻点是

更多咨询请联系16621398022（同微信）