📚 Mastering Differentiation for OxfordAQA International A Level Mathematics 9660 | 掌握 OxfordAQA 国际 A Level 数学 9660 微分
Differentiation is one of the most heavily examined topics in the OxfordAQA International A Level Mathematics 9660 Pure Mathematics units. Topic tests often focus on both routine derivative calculations and the application of differentiation to gradients, tangents, stationary points, optimisation and modelling. This revision guide breaks down the key skills and exam strategies you need to master.
微分是 OxfordAQA 国际 A Level 数学 9660 纯数学模块中考查频率最高的主题之一。主题测试通常既考查常规的求导计算，也考查导数在斜率、切线、驻点、优化问题和数学建模中的应用。本复习指南将梳理你需要掌握的核心技能和考试策略。
1. Understanding the Derivative | 理解导数
ive measures the instantaneous rate of change of a function. If y = f(x), the derivative f' l be expected to interpret derivatives graphically and numerically, not just compute them.
导数衡量函数在某一点的瞬时变化率。若 y = f(x)，则导数 f'(x) 或 dy/dx 表示曲线在任意 x 处的切线斜率。在 OxfordAQA 9660 纯数学试卷中，你不仅要会计算导数，还要能从图形和数值角度解释导数的意义。
A positive derivative means the function is increasing at that point, while a negative derivative means it is decreasing. The magnitude of the derivative tells you how steep the curve is. A zero derivative indicates a horizontal tangent, which may correspond to a stationary point such as a maximum, minimum or point of inflection.
导数为正表示函数在该点递增，导数为负表示函数递减。导数的大小反映了曲线的陡峭程度。导数为零意味着切线水平，这可能对应驻点，如极大值点、极小值点或拐点。
2. Standard Derivatives and Rules | 标准导数与法则
You must know the derivatives of common functions by heart. These include powers, exponentials, logarithms and trigonometric functions. The table below summarises the standard results that appear constantly in topic tests.
你必须熟记常见函数的导数，包括幂函数、指数函数、对数函数和三角函数。下表总结了在主题测试中频繁出现的标准结果。
| f(x) | f'(x) |
|---|---|
| xⁿ | nxⁿ⁻¹ |
| eˣ | eˣ |
| ln x | 1/x |
| sin x | cos x |
| cos x | -sin x |
| tan x | sec² x |
The power rule applies to any real constant n, so it also works for roots and reciprocals if you rewrite them using indices. For example, √x can be written as x¹⁄² and 1/x² as x⁻² before differentiating.
幂法则适用于任意实常数 n，因此只要利用指数形式重写根式和倒数，幂法则同样适用。例如，√x 可以先写成 x¹⁄²，1/x² 可以先写成 x⁻²，然后再求导。
3. The Chain Rule | 链式法则
n the derivative of y with respect to x is the product of the derivative of y with respect to u and the derivative of u with respect to x.
链式法则适用于复合函数，即一个函数嵌套在另一个函数内部的情况。若 y = f(u) 且 u = g(x)，则 y 对 x 的导数等于 y 对 u 的导数与 u 对 x 的导数的乘积。
dy/dx = dy/du × du/dx
y = u⁵. Then dy/du = 5u⁴ and du/dx = 6x, giving dy/dx = 5(3x² + 1)⁴ × 6x = 30x(3x² + 1)⁴.
例如，要对 y = (3x² + 1)⁵ 求导，可令 u = 3x² + 1，则 y = u⁵。此时 dy/du = 5u⁴，du/dx = 6x，因此 dy/dx = 5(3x² + 1)⁴ × 6x = 30x(3x² + 1)⁴。
In many questions, the chain rule is combined with other rules, so always identify the outermost function first. Remember that the derivative of eᵏˣ is keᵏˣ and the derivative of ln(kx) is 1/x, where k is a constant.
在许多题目中，链式法则会与其他法则结合使用，因此要首先识别最外层的函数。记住 eᵏˣ 的导数是 keᵏˣ，而 ln(kx) 的导数是 1/x，其中 k 为常数。
4. Product and Quotient Rules | 乘积法则与商法则
e. The derivative is the first function times the derivative of the second, plus the second function times the derivative of the first.
当对两个函数的乘积 y = u v 求导时，应使用乘积法则。导数等于第一个函数乘以第二个函数的导数，再加上第二个函数乘以第一个函数的导数。
dy/dx = u dv/dx + v du/dx
le applies. The derivative is the denominator times the derivative of the numerator, minus the numerator times the derivative of the denominator, all divided by the square of the denominator.
对于商 y = u/v，则使用商法则。导数等于分母乘以分子的导数，减去分子乘以分母的导数，再除以分母的平方。
dy/dx = (v du/dx - u dv/dx) / v²
Always write out u, v, du/dx and dv/dx separately before substituting into these formulas. This reduces sign errors and helps you earn method marks even if a small simplification slips later.
在代入公式之前，务必先单独写出 u、v、du/dx 和 dv/dx。这样可以减少符号错误，并且即使后续化简出现小失误，也能帮助你获得过程分。
5. Second Derivatives and Concavity | 二阶导数与凹凸性
The second derivative f''(x) or d²y/dx² is the derivative of f'(x). It describes the rate of change of the gradient and tells you whether a curve is bending upwards or downwards. A positive second derivative means the curve is concave upward or convex, while a negative second derivative means the curve is concave downward.
二阶导数 f''(x) 或 d²y/dx² 是 f'(x) 的导数。它描述斜率的变化率，并告诉你曲线的弯曲方向是向上还是向下。二阶导数为正表示曲线向上凹（凸），为负表示曲线向下凹。
To find the second derivative, simply differentiate f' nd d²y/dx² = 6x - 12. The sign of d²y/dx² at a stationary point helps classify its nature.
要求二阶导数，只需对 f'(x) 再求一次导。例如，若 y = x³ - 6x² + 9x，则 dy/dx = 3x² - 12x + 9，d²y/dx² = 6x - 12。驻点处 d²y/dx² 的符号有助于判断该点的性质。
6. Tangents and Normals | 切线与法线
The gradient of the tangent to a curve y = f(x) at the point where x = a is given by m = f' icular to the tangent, so its gradient is the negative reciprocal of the tangent gradient.
曲线 y = f(x) 在 x = a 处切线的斜率由 m = f'(a) 给出。切线方程可使用点斜式写出。法线垂直于切线，因此法线的斜率是切线斜率的负倒数。
Tangent: y - f(a) = f'(a)(x - a)
Normal: y - f(a) = -1/f'(a)(x - a)
If f' ative is undefined at a point, use limits or graphical reasoning to interpret the tangent.
如果 f'(a) = 0，则切线水平，法线竖直，此时法线方程应写为 x = a。如果某点处导数不存在，则需要利用极限或图形推理来判断切线。
7. Stationary Points and Curve Sketching | 驻点与曲线草图
Stationary points occur where f'(x) = 0. To classify a stationary point, you can either examine the sign of f'(x) on either side of the point or evaluate the second derivative f''(x). A local maximum has f'(x) changing from positive to negative, while a local minimum has f'(x) changing from negative to positive.
驻点出现在 f'(x) = 0 处。要判断驻点的性质，可以考察该点两侧 f'(x) 的符号，也可以计算二阶导数 f''(x)。局部极大值处 f'(x) 由正变负，局部极小值处 f'(x) 由负变正。
If f'(x) does not change sign, the stationary point is a point of inflection with a horizontal tangent. If f''(x) > 0 at a stationary point, it is a local minimum; if f''(x) < 0, it is a local maximum.
如果 f'(x) 不变号，则该驻点为水平切线的拐点。若驻点处 f''(x) > 0，则为局部极小值；若 f''(x) < 0，则为局部极大值。
Curve sketching questions often require you to find intercepts, stationary points and asymptotic behaviour. A clear table showing the sign of f'(x) in each interval helps you present accurate shape information.
曲线草图题通常要求找出截距、驻点和渐近行为。用清晰的表格展示各区间内 f'(x) 的符号，有助于准确呈现曲线形状信息。
8. Optimisation Problems | 优化问题
Many topic tests include a modelling question where you must express a quantity such as area, volume or profit as a function of one variable, then differentiate to find its maximum or minimum. Always identify the quantity to be optimised and write it in terms of a single variable using given constraints.
许多主题测试包含建模题：你需要将面积、体积或利润等量表示为一个变量的函数，然后通过求导找到其最大值或最小值。务必先确定要优化的量，并利用给定约束将其写成单一变量的函数。
After finding the stationary points, confirm which one gives the desired optimum by using the second derivative test or by checking endpoint values. State your conclusion clearly in the context of the problem, including units where appropriate.
找到驻点后，通过二阶导数检验或检查端点值来确认哪个驻点给出所需的最优值。请结合问题背景清晰地陈述结论，并在适当时包含单位。
9. Parametric Differentiation | 参数方程微分
When x and y are both given as functions of a parameter t, you can find the gradient dy/dx without eliminating the parameter. Differentiate x and y with respect to t and then divide the two derivatives.
当 x 和 y 都表示为参数 t 的函数时，无需消去参数即可求出斜率 dy/dx。分别对 x 和 y 关于 t 求导，然后将两个导数相除。
dy/dx = (dy/dt) / (dx/dt)
The second derivative is obtained by differentiating dy/dx with respect to t and then dividing by dx/dt. This is a common source of errors, so do not forget the final division by dx/dt.
二阶导数通过对 dy/dx 关于 t 求导后再除以 dx/dt 得到。这是常见的出错点，不要忘记最后除以 dx/dt。
d²y/dx² = [d/dt(dy/dx)] / (dx/dt)
= (3t² - 2) / 2t. Topic tests often ask for the tangent equation at a specific value of t.
例如，若 x = t² + 1，y = t³ - 2t，则 dx/dt
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