📚 Differentiation from First Principles and Applications | 从第一性原理到应用：微分核心技巧
Differentiation is one of the most important tools in A-Level Mathematics. It measures how a function changes as its input changes, giving the gradient of a curve at any point. In the Edexcel specification, candidates must be able to differentiate from first principles, use standard derivative rules, find tangents and normals, classify stationary points, and apply differentiation to real-world modelling problems. This revision guide works through every one of those skills in the order you meet them in the textbook, with the algebra shown in full so that you can copy the method into your own solutions.
微分是 A-Level 数学中最重要的工具之一。它衡量函数随输入变化而变化的速度，并给出曲线上任意一点的梯度。在 Edexcel 考试大纲中，考生必须掌握从第一性原理求导、使用标准求导法则、求切线与法线、判断驻点类型，以及将微分应用于实际建模问题。本复习指南按教材出现顺序逐一讲解这些技能，并完整展示代数推导过程，方便你把这些方法直接搬到自己的解题步骤中。
1. The Gradient of a Curve and the Difference Quotient | 曲线梯度与差商
lling there, and it changes from point to point. To estimate the gradient at x = a we take a nearby point x = a + h and calculate the average rate of change between the two points.
直线具有恒定梯度，但曲线不是。曲线 y = f(x) 在某一点的梯度衡量曲线在该处的陡峭程度，并且逐点变化。为了估计 x = a 处的梯度，我们取附近一点 x = a + h，并计算两点之间的平均变化率。
Average gradient = [ f(a + h) − f(a) ] ÷ h
aller, the chord gets closer to the tangent at x = a, so the average gradient approaches the true gradient of the curve at that point.
这个表达式称为差商，几何上它就是连接曲线上两点的弦的梯度。随着 h 越来越小，弦越来越接近 x = a 处的切线，因此平均梯度趋近于曲线在该点的真实梯度。
The key idea to remember is that a chord gives an estimate and a tangent gives the exact value. Differentiation is simply the process of turning the first into the second by taking a limit.
需要牢记的核心思想是：弦给出估计值，切线给出精确值。微分本质上就是通过取极限，把前者变成后者的过程。
2. Differentiation from First Principles | 从第一性原理出发求导
The derivative of f(x) is defined as the limit of the difference quotient as h tends to 0. This definition is the foundation of the whole topic, and Edexcel papers may ask you to prove the derivative of a simple function such as x², x³ or 1/x using it.
f(x) 的导数定义为差商在 h 趋向于 0 时的极限。这个定义是整个专题的基础，Edexcel 试卷可能会要求你用它来证明 x²、x³ 或 1/x 等简单函数的导数。
f '(x) = lim (h → 0) [ f(x + h) − f(x) ] ÷ h
An equivalent version uses a point a instead of x and lets the difference shrink towards zero:
等价的写法是用点 a 代替 x，并让差值趋向于零：
f '(a) = lim (x → a) [ f(x) − f(a) ] ÷ (x − a)
hen let h → 0. Any surviving term that still has an h in it disappears, leaving only the terms without h.
标准的四步法是：写出 f(x + h)；减去 f(x)；除以 h 并化简，约去所有含有 h 因子的项；最后令 h → 0。任何仍含 h 的项都会消失，只留下不含 h 的项。
x + h)² = x² + 2xh + h². Subtract f(x) = x², divide by h, and let h → 0. The result is 2x.
例如，若 f(x) = x²，则 f(x + h) = (x + h)² = x² + 2xh + h²。减去 f(x) = x²，再除以 h，并令 h → 0，结果为 2x。
3. Four Standard First-Principles Proofs | 四个标准第一性原理证明
and (x + h)³ = x³ + 3x²h + 3xh² + h³, subtract x³, divide by h to get 3x² + 3xh + h², then let h → 0 to obtain 3x².
考官反复使用的函数只有少数几个，因此值得把每个推导练到毫不犹豫就能写出。对于 f(x) = x³，展开 (x + h)³ = x³ + 3x²h + 3xh² + h³，减去 x³，除以 h 得到 3x² + 3xh + h²，再令 h → 0 得到 3x²。
ifies to −h ÷ [ x(x + h) ]. Dividing by h leaves −1 ÷ [ x(x + h) ], and as h → 0 this becomes −1 ÷ x².
对于 f(x) = 1/x，写出 f(x + h) − f(x) = 1 ÷ (x + h) − 1 ÷ x。通分到公分母 x(x + h)，得到 [ x − (x + h) ] ÷ [ x(x + h) ]，化简为 −h ÷ [ x(x + h) ]。再除以 h 得 −1 ÷ [ x(x + h) ]，当 h → 0 时变为 −1 ÷ x²。
For f(x) = √x, multiply numerator and denominator by the conjugate √x + h + √x. The numerator becomes h and cancels with the h underneath, leaving 1 ÷ [ √x + h + √x ], which tends to 1 ÷ (2√x).
对于 f(x) = √x，用共轭式 √x + h + √x 同时乘分子和分母。分子变成 h，与分母中的 h 相消，剩下 1 ÷ [ √x + h + √x ]，其极限为 1 ÷ (2√x)。
| f(x) | f '(x) from first principles | 关键化简技巧 |
| x² | 2x | 展开平方，消去 x² |
| x³ | 3x² | 展开立方，消去 x³ |
| 1 ÷ x | −1 ÷ x² | 通分成单一分数 |
| √x | 1 ÷ (2√x) | 乘以共轭式有理化 |
4. Notation and the Standard Rules of Differentiation | 记号与标准求导法则
Two notations are used side by side. Leibniz notation writes dy ÷ dx and reminds you that the derivative is a rate of change of y with respect to x. Lagrange notation writes f '(x) and is more convenient when you want to evaluate the derivative at a specific point, for example f '(3). The second derivative is written d²y ÷ dx² or f ''(x).
两种记号会同时使用。莱布尼茨记号写作 dy ÷ dx，提醒你导数是 y 关于 x 的变化率。拉格朗日记号写作 f '(x)，当你想在特定点求导数值（如 f '(3)）时更方便。二阶导数写作 d²y ÷ dx² 或 f ''(x)。
positive integers, negative integers and fractions, which means it also covers 1 ÷ x² and √x without any extra work. The constant multiple rule lets you pull a number out of the derivative, and the sum rule lets you differentiate term by term.
幂法则指出，若 y = xⁿ，则 dy ÷ dx = n xⁿ⁻¹。它对正整数、负整数和分数都成立，因此无需额外工作就能处理 1 ÷ x² 和 √x。常数倍法则允许把数字提到导数外面，和法则允许逐项求导。
d ÷ dx ( a xⁿ ) = a n xⁿ⁻¹
A useful exam habit is to rewrite every root and reciprocal as a power of x before differentiating. For example 3 ÷ x⁴ becomes 3x⁻⁴, whose derivative is −12x⁻⁵, that is −12 ÷ x⁵.
一个实用的考试习惯是：在求导之前，先把所有根式和倒数改写为 x 的幂。例如 3 ÷ x⁴ 可写成 3x⁻⁴，其导数为 −12x⁻⁵，即 −12 ÷ x⁵。
5. Tangents and Normals | 切线与法线
t at that point. So to find the tangent to y = f(x) at x = a, first differentiate to get f '(x), then substitute x = a to get the gradient m = f '(a), and finally use the straight-line formula with the point (a, f(a)).
导数在某点的值正是该点切线的梯度。因此，求 y = f(x) 在 x = a 处的切线时，先求导得 f '(x)，再代入 x = a 得到梯度 m = f '(a)，最后用点 (a, f(a)) 套用直线方程公式。
y − y₁ = m ( x − x₁ )
The normal is the line perpendicular to the tangent at the point of contact. If the tangent has gradient m, the normal has gradient −1 ÷ m, provided m is not zero. A horizontal tangent therefore has a vertical normal, and vice versa.
法线是在切点处与切线垂直的直线。若切线梯度为 m，则法线梯度为 −1 ÷ m（m 不为零时）。因此水平切线对应竖直法线，反之亦然。
Exam questions often ask you to show that a tangent or normal passes through a given point, or to find where the normal meets the x-axis. Always leave your gradient as an exact fraction rather than a rounded decimal, because the follow-up parts usually depend on the exact value.
考题常要求你证明某条切线或法线经过给定点，或求法线与 x 轴的交点。务必把梯度写成精确分数而不是四舍五入的小数，因为后续小题通常依赖这个精确值。
6. Stationary Points and the Second Derivative Test | 驻点与二阶导数判别法
e x-coordinates; you must then substitute back into the original equation, not the derivative, to get the y-coordinates.
驻点是梯度为零的点，因此通过解 dy ÷ dx = 0 求得。解出的只是横坐标；你必须把它们代回原方程（而不是导数）求出纵坐标。
To classify each stationary point, evaluate the second derivative there. If d²y ÷ dx² is positive the point is a local minimum, and if it is negative the point is a local maximum. If it equals zero the test fails and you must examine the sign of the gradient just to the left and just to the right of the point instead.
判断每个驻点的类型，需要在驻点处求二阶导数值。若 d²y ÷ dx² 为正，则该点是局部极小值点；若为负，则是局部极大值点。若等于零，判别法失效，此时必须改为考察该点左侧和右侧梯度的符号。
minimum d²y ÷ dx² < 0 → maximum d²y ÷ dx² = 0 → inconclusive
A common loss of marks is to state the nature of a stationary point without any justification. Write the second derivative explicitly, substitute the x-value, and state the numerical result before drawing the conclusion.
常见失分点是只给出驻点类型而不作论证。请明确写出二阶导数，代入横坐标，给出数值结果，然后再下结论。
7. Increasing, Decreasing Functions and Concavity | 增减性与凹凸性
A function is increasing on an interval where f '(x) > 0 and decreasing where f '(x) < 0. To answer an exam question of this type, differentiate, solve the inequality, and present the answer as an interval of x-values rather than a single point.
函数在 f '(x) > 0 的区间上递增，在 f '(x) < 0 的区间上递减。回答这类考题时，先求导，再解不等式，并把答案写成 x 的区间而不是单个点。
The second derivative also describes concavity. Where d²y ÷ dx² > 0 the curve is convex, sometimes called concave upwards, and it sits above its tangents. Where d²y ÷ dx² < 0 the curve is concave, and it sits below its tangents. A point where the concavity changes is called a point of inflection, and at such a point the second derivative is zero while the first derivative need not be.
二阶导数还描述凹凸性。当 d²y ÷ dx² > 0 时曲线为凸（也称下凹），位于其切线之上；当 d²y ÷ dx² < 0 时曲线为凹，位于其切线之下。凹凸性发生改变的点称为拐点，在拐点处二阶导数为零，而一阶导数不必为零。
Note the important distinction: a stationary point is about the first derivative, an inflection point is about the second derivative. A point can be one, the other, both or neither, and marking schemes reward candidates who keep the two ideas separate.
请注意重要区别：驻点关乎一阶导数，拐点关乎二阶导数。一个点可以是其中之一、两者兼具或两者皆非，评分标准会奖励能把两个概念区分清楚的考生。
8. Chain, Product and Quotient Rules | 链式、乘积与商法则
Composite functions are differentiated with the chain rule. If y is a function of u and u is a function of x, then the rates of change multiply.
复合函数用链式法则求导。若 y 是 u 的函数，u 又是 x 的函数，则变化率相乘。
dy ÷ dx = ( dy ÷ du ) × ( du ÷ dx )
In function notation the chain rule says that the derivative of f(g(x)) is f '(g(x)) × g' derivative of the inside, so the derivative of (3x + 1)⁵ is 5(3x + 1)⁴ × 3 = 15(3x + 1)⁴.
用函数记号表示，链式法则说明 f(g(x)) 的导数是 f '(g(x)) × g'(x)。大多数考生使用的实用版本是：对外层函数求导、内层保持不变，再乘以内层的导数，因此 (3x + 1)⁵ 的导数为 5(3x + 1)⁴ × 3 = 15(3x + 1)⁴。
Products and quotients have their own rules, and mixing them up is one of the most frequent errors in the whole paper.
乘积与商各有自己的法则，而把它们混淆是整份试卷中最常见的错误之一。
( u v )' = u ' v + u v ' | ( u ÷ v )' = ( u ' v − u v ' ) ÷ v²
Notice that the product rule keeps a plus sign, while the quotient rule has a minus sign and a squared denominator. In the quotient rule the numerator always begins with the derivative of the top multiplied by the bottom, and the order of the subtraction matters, because reversing it changes the sign of the whole answer.
注意乘积法则用加号，而商法则用减号且分母平方。商法则的分子永远以「分子的导数乘分母」开头，减法的顺序很重要，颠倒顺序会改变整个答案的符号。
9. Implicit and Parametric Differentiation | 隐函数与参数方程求导
e a term in y you must multiply by dy ÷ dx because of the chain rule. For the circle x² + y² = 25 this gives 2x + 2y ( dy ÷ dx ) = 0, so dy ÷ dx = −x ÷ y.
有些曲线以隐函数形式给出，y 并未单独解出。此时对每一项关于 x 求导，并记住：对含 y 的项求导时必须因链式法则而乘以 dy ÷ dx。对于圆 x² + y² = 25，得到 2x + 2y ( dy ÷ dx ) = 0，故 dy ÷ dx = −x ÷ y。
This result is worth checking geometrically: the gradient of the tangent to a circle is perpendicular to the radius, and −x ÷ y is indeed the negative reciprocal of the radius gradient y ÷ x.
这个结果值得从几何角度检验：圆的切线梯度与半径垂直，而 −x ÷ y 正是半径梯度 y ÷ x 的负倒数。
Parametric curves are handled separately. If x and y are both given in terms of a parameter t, the gradient of the curve is the ratio of the two derivatives, with dt cancelling.
参数方程曲线需单独处理。若 x 与 y 都以参数 t 表示，则曲线的梯度是两个导数之比，dt 相互抵消。
dy ÷ dx = ( dy ÷ dt ) ÷ ( dx ÷ dt )
Two warnings are worth repeating. First, dy ÷ dx is not the same as dy ÷ dt, so never quote a parametric derivative without dividing. Second, the second derivative of a parametric curve is not the simple quotient of the two second derivatives; you must differentiate dy ÷ dx with respect to t and then divide by dx ÷ dt again.
有两点提醒值得重复。第一，dy ÷ dx 与 dy ÷ dt 不同，因此绝不要未作除法就给出参数方程的导数。第二，参数曲线的二阶导数并不是两个二阶导数的简单商；你必须先对 dy ÷ dx 关于 t 求导，再除以 dx ÷ dt。
10. Exponential, Logarithmic and Trigonometric Derivatives | 指数、对数与三角函数的导数
The exponential function is the special function whose derivative equals itself. This self-reproducing property is why e appears everywhere in growth and decay models. The natural logarithm is its inverse, and its derivative is the reciprocal function.
指数函数是导数等于自身的特殊函数。正是这种「自我复制」性质使 e 频繁出现在增长与衰减模型中。自然对数是它的反函数，其导数为倒数函数。
d ÷ dx ( eˣ ) = eˣ | d ÷ dx ( ln x ) = 1 ÷ x | d ÷ dx ( aˣ ) = aˣ ln a
The trigonometric derivatives form a cycle. Sine differentiates to cosine, cosine to negative sine, and tangent to sec²x, which is also written 1 ÷ cos²x.
三角函数的导数构成一个循环。sin 求导得 cos，cos 求导得 −sin，tan 求导得 sec²x，也可写作 1 ÷ cos²x。
d ÷ dx ( sin x ) = cos x | d ÷ dx ( cos x ) = −sin x | d ÷ dx ( tan x ) = sec²x
Each of these has a chained version that appears constantly in applied questions. The derivative of sin(kx) is k cos(kx), the derivative of cos(kx) is −k sin(kx), and the derivative of ekx is k ekx. Likewise ln(kx) differentiates to 1 ÷ x, since the constant k only shifts the graph vertically.
这些公式各有一个链式版本，在应用题中频繁出现。sin(kx) 的导数是 k cos(kx)，cos(kx) 的导数是 −k sin(kx)，ekx 的导数是 k ekx。同样，ln(kx) 的导数为 1 ÷ x，因为常数 k 只是把图像上下平移。
11. Optimisation, Rates of Change and Modelling | 最优化、变化率与建模应用
Applied differentiation questions follow a reliable structure. Define your variables and write down what you are trying to maximise or minimise. Form an equation linking the quantities, using a constraint to eliminate one variable so that the target becomes a function of a single variable. Differentiate, set the derivative to zero, solve, and justify the nature of the answer with the second derivative or a sign argument.
微分应用题遵循可靠的结构。先定义变量，写出你要求最大值或最小值的量。建立联系各量的方程，利用约束条件消去一个变量，使目标量成为单变量函数。然后求导、令导数为零、求解，并用二阶导数或符号论证来验证答案的性质。
A classic example is a closed rectangular box made from a fixed area of card. If the base is a square of side x and the height is y, the surface area 2x² + 4xy is fixed, so y can be expressed in terms of x. The volume x²y then becomes a function of x alone, and differentiating and setting to zero gives the side length that maximises the volume.
经典例子是用固定面积的纸板制作密闭长方体盒。若底面为边长 x 的正方形、高为 y，则表面积 2x² + 4xy 固定，于是 y 可用 x 表示。体积 x²y 就变成只含 x 的函数，求导并令其为零即可得到使体积最大的边长。
Connected rates of change questions link two derivatives through the chain rule. If the radius of a balloon grows at a known rate, the rate of change of volume is found by writing V in terms of r, differentiating to get dV ÷ dr, and multiplying by dr ÷ dt. Always state units in your final answer, since a rate without units is incomplete.
相关变化率问题通过链式法则联系两个导数。若气球半径以已知速率增长，则体积的变化率可由 V 关于 r 的表达式求出 dV ÷ dr，再乘以 dr ÷ dt 得到。最终答案务必写出单位，因为缺少单位的变化率是不完整的。
dV ÷ dt = ( dV ÷ dr ) × ( dr ÷ dt )
Finally, remember that any answer to a modelling question must be interpreted in context. If the variable is a length or a time, a negative root of your equation should be rejected, and any value outside the physical domain of the problem should be discarded.
最后，请记住建模题的答案必须回到实际情境中解释。若变量是长度或时间，方程中出现的负根应当舍去，任何超出问题实际取值范围的解也应剔除。
12. Exam Technique and Common Errors | 考试技巧与常见错误
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