📚 Prelims 7H: Mastering Harder Differentiation Problems | 预考 7H：攻克高等微分难题
In A-level Mathematics, preliminary examination papers often include a question labelled 7H that tests deeper applications of differentiation. This article walks through the core skills, common pitfalls, and exam strategies you need for such harder differentiation questions.
在 A-level 数学预考试卷中，常有一道编号为 7H 的题目，专门考查微分的深层应用。本文将带你系统梳理解决此类高等微分题所需的核心技能、常见错误与应试策略。
1. Understanding the Derivative from First Principles | 从第一原理理解导数
The derivative measures the instantaneous rate of change of a function. From first principles, it is defined as the limit of the difference quotient as h approaches zero.
导数度量函数在某一点的瞬时变化率。从第一原理出发，导数定义为差商的极限，其中 h 趋于 0。
f'(x) = lim (h → 0) [f(x + h) - f(x)] / h
In a prelim question, you may be asked to prove the derivative of xⁿ or sin x using this limit. Manipulating the numerator and applying standard limits is usually required.
在预考题中，可能要求你用该极限证明 xⁿ 或 sin x 的导数。通常需要整理分子并应用标准极限。
For example, expanding (x + h)ⁿ by the binomial theorem and then dividing by h shows that the derivative of xⁿ is nxⁿ⁻¹ for positive integer n.
例如，用二项式定理展开 (x + h)ⁿ，再除以 h，就能证明 xⁿ 的导数为 nxⁿ⁻¹（n 为正整数）。
2. Rules of Differentiation: Product, Quotient and Chain | 微分法则：乘积、商与链式法则
Harder differentiation problems rarely involve a single basic function. You must be fluent with the product rule, quotient rule, and chain rule.
较难的微分题很少只涉及单一基本函数。你必须熟练运用乘法法则、除法法则和链式法则。
| Product rule | (uv)' = u'v + uv' |
| Quotient rule | (u/v)' = (u'v - uv') / v² |
| Chain rule | dy/dx = dy/du × du/dx |
For instance, to differentiate y = e²ˣ sin 3x, apply the product rule together with the chain rule: y' = 2e²ˣ sin 3x + 3e²ˣ cos 3x.
例如，对 y = e²ˣ sin 3x 求导时，需要同时使用乘法法则和链式法则：y' = 2e²ˣ sin 3x + 3e²ˣ cos 3x。
In many prelim questions, identifying which rule to apply first is as important as applying it correctly. Simplify algebraic expressions before differentiating where possible.
在许多预考题中，判断先使用哪一条法则与正确使用法则同样重要。只要可能，在求导前先化简代数表达式。
3. Tangents and Normals: Equations and Geometrical Meaning | 切线与法线：方程及几何意义
The gradient of the tangent at x = a is f'(a). The tangent line is y - f(a) = f'(a)(x - a). The normal has gradient -1/f'(a) provided f'(a) ≠ 0.
在 x = a 处切线的斜率为 f'(a)。切线方程为 y - f(a) = f'(a)(x - a)。法线斜率为 -1/f'(a)（假设 f'(a) 不等于 0）。
A common prelim trap is to give the normal equation but ask for the tangent, or to require implicit differentiation before evaluating the gradient.
预考中常见的陷阱是给出法线方程却要求切线，或者需要先隐函数求导再计算斜率。
s vertical and has equation x = a. When the tangent is vertical, the normal is horizontal.
当切线水平时，其斜率为零，因此法线竖直，方程为 x = a。当切线竖直时，法线水平。
4. Stationary Points: Turning Points and Points of Inflection | 驻点：转折点与拐点
Stationary points occur where f' mum or minimum; a stationary point of inflection has zero gradient but does not change from increasing to decreasing.
驻点出现在 f'(x) = 0 处。转折点是局部极大值或极小值；驻点拐点的梯度为零，但函数并不由递增转为递减。
To classify a stationary point, use either the first derivative test (sign change of f'(x)) or the second derivative test (f''(x) > 0 for minimum, f''(x) < 0 for maximum, f''(x) = 0 inconclusive).
判别驻点类型可使用一阶导数判别法（观察 f'(x) 的符号变化）或二阶导数判别法（f''(x) > 0 为极小值，f''(x) < 0 为极大值，f''(x) = 0 则无法确定）。
If the second derivative test is inconclusive, examine the sign of f'(x) on either side of the stationary point to decide its nature.
如果二阶导数判别法无法确定，可检查驻点两侧 f'(x) 的符号来判断其性质。
5. Increasing and Decreasing Functions | 增函数与减函数
A function is increasing on an interval if f'(x) > 0 throughout that interval, and decreasing if f'(x) < 0. For a rational or radical function, you must also consider the domain.
若在某一区间内 f'(x) > 0，则函数在该区间递增；若 f'(x) < 0 则递减。对于有理函数或根式函数，必须同时考虑定义域。
Prelim questions often ask for the set of values of x for which the function is increasing, requiring you to solve a quadratic or rational inequality.
预考题常要求求出函数递增时 x 的取值范围，需要解二次不等式或有理不等式。
For example, if f' (x - 1)(x + 2), then f is increasing when x < -2 or x > 1, but decreasing when -2 < x < 1.
例如，若 f'(x) = (x - 1)(x + 2)，则函数在 x < -2 或 x > 1 时递增，在 -2 < x < 1 时递减。
6. The Second Derivative and Concavity | 二阶导数与凹凸性
The second derivative f''(x) tells you how the gradient changes. If f''(x) > 0, the curve is concave upward; if f''(x) < 0, it is concave downward.
二阶导数 f''(x) 告诉你斜率如何变化。若 f''(x) > 0，曲线向上凹；若 f''(x) < 0，曲线向下凹。
A point of inflection occurs where the concavity changes, usually where f''(x) = 0 and f''(x) changes sign.
拐点出现在凹凸性改变处，通常在 f''(x) = 0 且 f''(x) 改变符号的位置。
Not every point with f''(x) = 0 is an inflection point; you must verify the sign change of the second derivative.
并非所有 f''(x) = 0 的点都是拐点；必须验证二阶导数符号发生改变。
7. Rates of Change: Connected Rates Problems | 变化率：相关变化率问题
rule gives dy/dt = (dy/dx)(dx/dt). You must identify the known rate and the required rate.
如果 y 是 x 的函数且二者都随时间 t 变化，链式法则给出 dy/dt = (dy/dx)(dx/dt)。你必须识别已知变化率和所求变化率。
In geometrical contexts, differentiate a formula such as volume or area with respect
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