Mixed Exercise 9: Differentiation Masterclass | 混合练习9：微分专题精讲

📚 Mixed Exercise 9: Differentiation Masterclass | 混合练习9：微分专题精讲

Chapter 9 of Edexcel Pure Mathematics Year 2 consolidates the differentiation of trigonometric, exponential and logarithmic functions, and introduces the chain rule, the product rule and the quotient rule. This article mirrors the structure of Mixed Exercise 9, presenting revision notes and worked-style examples that target the most commonly tested skills in the A-Level examination.

Edexcel 纯数学 Year 2 第 9 章系统讲解了三角、指数与对数函数的微分法则，并引入了链式法则、乘积法则与商法则。本文紧随《混合练习 9》的题型结构，为你梳理高频考点、核心方法与易错警示，直击 A-Level 考试中最常见的考查方式。

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1. Core Standard Derivatives | 基本导数公式回顾

be automatic. If y = sin x then dy/dx = cos x; if y = cos x then dy/dx = -sin x; if y = eˣ then dy/dx = eˣ exactly, with signs and domains correct, because every later rule is built on top of them.

在进入混合练习之前，四个基本求导结果必须做到条件反射。若 y = sin x，则 dy/dx = cos x；若 y = cos x，则 dy/dx = -sin x；若 y = eˣ，则 dy/dx = eˣ；若 y = ln x，则 dy/dx = 1/x（x > 0）。这些公式必须精确记忆，尤其是符号与定义域不能出错，因为后面所有法则都建立在它们之上。

d/dx (sin x) = cos x   d/dx (cos x) = -sin x   d/dx (eˣ) = eˣ   d/dx (ln x) = 1/x

end to composite arguments: d/dx[sin(kx)] = k cos(kx), d/dx[cos(kx)] = -k sin(kx), d/dx[eᵏˣ] = k eᵏˣ sult is independent of the constant k, which surprises many students in mock examinations.

一旦掌握链式法则，这些公式就能推广到复合变量：d/dx[sin(kx)] = k cos(kx)，d/dx[cos(kx)] = -k sin(kx)，d/dx[eᵏˣ] = k eᵏˣ，而 d/dx[ln(kx)] = 1/x。注意最后一个结果与常数 k 无关，这一点常令许多学生在模考中感到意外。

y	dy/dx	Key reminder
sin x	cos x	No sign change
cos x	-sin x	The minus sign is the classic slip
eˣ	eˣ	Unchanged by differentiation
ln x	1/x	Valid only for x > 0

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2. Differentiating Trigonometric Functions | 三角函数微分

Exam questions rarely stop at plain sin x. The trigonometric expressions in Mixed Exercise 9 usually arrive wrapped inside a composite, a product or a quotient, and your job is to identify which structure you are looking at before writing anything down.

考试题很少停留在纯粹的 sin x 上。《混合练习 9》中的三角表达式通常嵌套在复合、乘积或商的结构里，你的任务是先判断结构，再动笔计算。

• x² + 1) → apply the chain rule and multiply by the derivative of the inside function.

• 三角函数内部是复合式，例如 y = sin(3x² + 1)，需用链式法则，再乘以内部函数的导数。

• ction, such as y = cos²x = (cos x)² → treat it as a chain with the outer function u².

• 三角函数的幂，例如 y = cos²x = (cos x)²，把它看作外层函数 u² 的复合结构。

• cts of a polynomial and a trig function, such as y = x² sin x → use the product rule.

• 多项式与三角函数的乘积，例如 y = x² sin x，使用乘积法则。

• ents such as y = sin x / x → use the quotient rule, or rewrite with a negative index.

• 商的形式，例如 y = sin x / x，使用商法则，或改写为负指数形式。

then dy/dx = 2x sin x + x² cos x; and if y = sin x / x then dy/dx = (x cos x - sin x)/x².

需要烂熟于心的示范：若 y = sin(3x + 1)，则 dy/dx = 3 cos(3x + 1)；若 y = cos²x，则 dy/dx = 2 cos x × (-sin x) = -2 sin x cos x = -sin 2x；若 y = x² sin x，则 dy/dx = 2x sin x + x² cos x；若 y = sin x / x，则 dy/dx = (x cos x - sin x)/x²。

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3. The Chain Rule | 链式法则

The chain rule handles any function of a function. If y is a function of u and u is a function of x, then the derivatives multiply together, and the whole skill lies in choosing u well - usually the expression inside the brackets, under the root, or in the exponent.

链式法则处理一切"函数中的函数"。若 y 是 u 的函数，而 u 是 x 的函数，则两个导数相乘；其中的技巧全在于如何合理选取 u——通常取括号内、根号下或指数上的那个表达式。

dy/dx = (dy/du) × (du/dx)

Typical applications in Mixed Exercise 9 include y = e^{3x² + 1}, where dy/dx = 6x e^{3x² + 1} y = ln(5x - 2), where dy/dx = 5/(5x - 2); y = sin³x, where dy/dx = 3 sin²x cos x; and y = √1 + x², where dy/dx = x/√1 + x².

《混合练习 9》中的典型应用包括：y = e^{3x² + 1}，则 dy/dx = 6x e^{3x² + 1}；y = ln(5x - 2)，则 dy/dx = 5/(5x - 2)；y = sin³x，则 dy/dx = 3 sin²x cos x；y = √1 + x²，则 dy/dx = x/√1 + x²。

Two warnings. First, never forget the inner derivative - losing it is the single most expensive error in this chapter. Second, when the answer can be simplified (for example turning 2 sin x cos x into sin 2x), simplify, because mark schemes often award a final accuracy mark only for the neatest form.

两点警告。第一，绝不要忘记内部导数，漏乘它是本章代价最大的错误。第二，若答案可以化简（例如把 2 sin x cos x 写成 sin 2x），一定要化简，因为评分标准常常只对最简形式给出最后的分值。

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4. The Product Rule | 乘积法则

The product rule applies when two differentiable expressions are multiplied. Write down u and v explicitly, differentiate each one separately, then substitute into the formula rather than trying to do it mentally.

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当两个可导表达式相乘时使用乘积法则。先把 u 与 v 明确写出，分别求导，再代入公式，不要试图心算完成。

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

A common mnemonic is "first times derivative of second, plus second times derivative of first". The order of the two terms does not matter for the final answer, but writing them in a consistent order keeps your working readable and reduces algebraic slips under exam pressure.

常用口诀是"前导后不导，加后导前不导"，即第一项乘第二项的导数，加上第二项乘第一项的导数。两项顺序不影响最终结果，但保持一致的书写顺序能让过程清晰，减少考试压力下的代数失误。

Worked examples: if y = x² eˣ then dy/dx = 2x eˣ + x² eˣ = eˣ n x then dy/dx = ln x + 1; if y = x³ cos x then dy/dx = 3x² cos x - x³ sin x; and if y = eˣ sin x then dy/dx = eˣ sin x + eˣ cos x = eˣ(sin x + cos x).

示范例题：若 y = x² eˣ，则 dy/dx = 2x eˣ + x² eˣ = eˣ(x² + 2x)；若 y = x ln x，则 dy/dx = ln x + 1；若 y = x³ cos x，则 dy/dx = 3x² cos x - x³ sin x；若 y = eˣ sin x，则 dy/dx = eˣ sin x + eˣ cos x = eˣ(sin x + cos x)。

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5. The Quotient Rule | 商法则

The quotient rule is needed whenever one function is divided by another and the division cannot be turned into a product easily. As with the product rule, label the numerator u and the denominator v before you start.

当两个函数相除、且难以改写为乘积时，就需要商法则。与乘积法则一样，动笔前先把分子记为 u、分母记为 v。

d/dx (u/v) = [ v (du/dx) - u (dv/dx) ] / v²

The most memorable phrasing is "denominator times derivative of numerator minus numerator times derivative of denominator, all over denominator squared". The subtraction order is not optional: reversing it flips the sign of the entire answer and typically costs two or three marks.

最好记的说法是"分母乘分子的导数，减去分子乘分母的导数，整体除以分母的平方"。减法的顺序不可颠倒：一旦写反，整道题的答案符号全错，通常会丢掉两到三分。

+ 1)(1)]/(x - 3)² = -7/(x - 3)²; if y = ln x / x then dy/dx = (1 - ln x)/x²; and if y = eˣ/(x + 1) then dy/dx = x eˣ/(x + 1)².

示范例题：若 y = (2x + 1)/(x - 3)，则 dy/dx = [(x - 3)(2) - (2x + 1)(1)]/(x - 3)² = -7/(x - 3)²；若 y = ln x / x，则 dy/dx = (1 - ln x)/x²；若 y = eˣ/(x + 1)，则 dy/dx = x eˣ/(x + 1)²。

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6. Exponential and Logarithmic Functions | 指数与对数函数微分

e derivative is the original function multiplied by the derivative of the exponent: d/dx[e^f(x)] = f'(x) e^f vative is the derivative of the argument divided by the argument itself: d/dx[ln f(x)] = f'(x)/f(x).

对于复合指数函数，导数为原函数乘以指数的导数：d/dx[e^f(x)] = f'(x) e^f(x)。对于复合对数函数，导数为内部函数的导数除以内部函数本身：d/dx[ln f(x)] = f'(x)/f(x)。

d/dx (e^f(x)) = f'(x) e^f(x)    d/dx (ln f(x)) = f'(x) / f(x)    f(x) > 0

n x for k > 0, its derivative is simply 1/x, and the constant k disappears. This is a favourite exam trap because students expect the constant to survive differentiation.

对数结果解释了前面提到的"意外"：因为当 k > 0 时 ln(kx) = ln k + ln x，其导数就是 1/x，常数 k 会消失。这是考试中最爱的陷阱之一，因为学生总以为常数一定会在求导后保留下来。

• y = e^5x → dy/dx = 5 e^5x

• y = e^-x² → dy/dx = -2x e^-x²

• y = ln(3x + 1) → dy/dx = 3/(3x + 1)

• y = ln(x² + 4) → dy/dx = 2x/(x² + 4)

• y = x ln x - x → dy/dx = ln x

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7. Differentiating aˣ and Bases Other Than e | 一般指数函数 aˣ 的微分

base is a positive constant a rather than e, rewrite the function using the identity a = e^{ln a}, so that aˣ = e^{x ln a} chain rule then produces the standard result, which is worth memorising as a single line.

当底数是正常数 a 而不是 e 时，利用恒等式 a = e^{ln a} 把函数改写为 aˣ = e^{x ln a}。再用链式法则即可得到标准结果，这一行结论值得直接背下来。

d/dx (aˣ) = aˣ ln a    (a > 0, a ≠ 1)

ppears. This is a useful consistency check: if your formula ever fails to reduce to d/dx(eˣ) = eˣ when a = e, something has gone wrong.

令 a = e 就回到熟悉的情形，因为 ln e = 1，多余的因子消失了。这是一个很好的一致性检验：若当 a = e 时你的公式无法退化为 d/dx(eˣ) =

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