📚 D - The quotient rule | D - 商的求导法则
In IB Mathematics, differentiation is a central topic, and once you have mastered the product rule, the next logical step is the quotient rule. The section labelled "D - The quotient rule" covers how to differentiate functions that are expressed as one function divided by another. This rule is essential for handling rational functions, trigonometric ratios like tan x, and many combinations involving exponentials and logarithms. Understanding the quotient rule will not only help you solve differentiation problems accurately but also deepen your appreciation of the structure of derivative rules.
在 IB 数学中，微分是一个核心主题，掌握了乘法法则之后，下一个自然的步骤就是商的求导法则。标为 “D - 商的求导法则” 的部分讲解如何对以一个函数除以另一个函数的形式给出的表达式求导。这一法则对于处理有理函数、三角比（如 tan x）以及许多涉及指数和对数的组合都是必不可少的。理解商法则不仅能帮助你准确求解微分问题，还能加深你对导数法则结构的认识。
1. What Is the Quotient Rule? | 什么是商法则？
n f(x) = u(x) / v(x), where both u and v are differentiable and v(x) ≠ 0, the derivative f ' nition of the derivative and can be derived using the product rule combined with the chain rule.
商法则是一个用来求两个可微函数之比的导数的公式。若函数 f(x) = u(x) / v(x)，其中 u 和 v 均可微且 v(x) ≠ 0，则导数 f '(x) 由特定法则给出，而不是简单地对分子分母分别求导。该法则源自导数的极限定义，也可利用乘法法则结合链式法则推导出来。
2. Statement of the Rule | 法则的表述
where u = u(x) and v = v(x) are functions of x. The derivative of y with respect to x is:
设 y = u / v，其中 u = u(x)、v = v(x) 均为 x 的函数。y 关于 x 的导数为：
dy/dx = (v du/dx − u dv/dx) / v²
In prime notation, if f(x) = u(x) / v(x), then f '(x) = (u' v − u v' f the bottom function. The denominator is simply the square of the original denominator. The order in the numerator is crucial: "v du/dx − u dv/dx", not the other way around.
用撇号记法，若 f(x) = u(x) / v(x)，则 f '(x) = (u' v − u v') / v²。注意分子是“上函数的导数乘以下函数，减去上函数乘以下函数的导数”。分母则是原分母的平方。分子中的顺序至关重要：“v du/dx − u dv/dx”，切不可颠倒。
3. Mnemonic Devices | 记忆技巧
A popular mnemonic to remember the quotient rule is: "low d high minus high d low, stroke the square of the low." Here, "low" refers to the denominator v, "high" refers to the numerator u, "d" means derivative, and "stroke the square" implies putting the square of the denominator underneath. Using Leibniz notation, you can think: "denominator times derivative of numerator minus numerator times derivative of denominator, all over denominator squared."
一个流行的记忆口诀是：“下乘上导减去上乘下导，除以下平方”。其中“下”指分母 v，“上”指分子 u，“导”表示导数。用莱布尼茨记法可以这样想：“分母乘以分子的导数减去分子乘以分母的导数，整体除以分母的平方”。
Another way to recall the pattern is to compare it with the product rule: for a product we add, but for a quotient we subtract and divide by the square of the denominator. Rehearsing the phrase "v u' − u v' over v²" while solving problems will engrave it in your memory.
另一种记忆方法是与乘法法则比较：乘法法则中间是加号，而商法则是减号并除以分母的平方。在解题时反复默念 “v u' − u v' 除以 v²”，能让你牢记于心。
4. Derivation Using the Product Rule | 利用乘法法则推导
· v⁻¹. Applying the product rule together with the chain rule to differentiate v⁻¹ yields:
通过将商 u/v 写为乘积 y = u · v⁻¹，可以推导出商法则。对 v⁻¹ 使用乘法法则和链式法则：
dy/dx = u' · v⁻¹ + u · (−1)v⁻² · v' = u'/v − u v'/v² = (u' v − u v') / v²
This derivation highlights why the denominator becomes squared and why a subtraction appears. It also reminds us that the quotient rule is essentially the product rule applied to u · (1/v). This connection is useful if you ever forget the formula, as you can always rewrite the quotient and apply the product and chain rules.
这个推导揭示了为何分母是平方，以及为何会出现减号。它也提醒我们，商法则本质上就是乘法法则应用于 u · (1/v)。如果忘记了公式，这一联系非常有用，因为你总可以将商改写为乘积，再应用乘法法则和链式法则。
5. Basic Examples | 基础示例
Consider f(x) = x / (x + 1). Here u = x, v = x + 1, so u' = 1, v' = 1. Plugging into the quotient rule:
考虑 f(x) = x / (x + 1)。此处 u = x，v = x + 1，因此 u' = 1，v' = 1。代入商法则：
f '(x) = (1·(x+1) − x·1) / (x+1)² = (x + 1 − x) / (x+1)² = 1 / (x+1)²
a second example, differentiate g(x) = (x² + 3) / (2x − 1). Here u = x² + 3, v = 2x − 1, u' = 2x, v' = 2. Therefore,
第二个例子，对 g(x) = (x² + 3) / (2x − 1) 求导。u = x² + 3，v = 2x − 1，u' = 2x，v' = 2。于是，
g ' 1) − (x²+3)·2) / (2x − 1)² = (4x² − 2x − 2x² − 6) / (2x − 1)² = (2x² − 2x − 6) / (2x − 1)²
Notice that simplifying the numerator often makes the final answer neater and easier to evaluate at specific points.
注意，化简分子后答案通常更整洁，也便于在某点处求值。
6. Trigonometric Functions | 三角函数
for the derivative of tan x. Since tan x = sin x / cos x, let u = sin x, v = cos x. Then u' = cos x, v' = −sin x. Applying the rule:
商法则为 tan x 的导数提供了简洁的证明。由于 tan x = sin x / cos x，令 u = sin x，v = cos x，则 u' = cos x，v' = −sin x。应用法则：
(cos x·cos x − sin x·(−sin x)) / cos² x = (cos² x + sin² x) / cos² x = 1 / cos² x = sec² x
. Mastering the quotient rule makes it simple to derive these results instead of memorizing them blindly.
同理，sec x = 1 / cos x 也可以用商法则求导（令 u = 1，v = cos x），得到 sec x tan x。掌握了商法则，就能够轻松推导这些结果，而不必死记硬背。
7. Exponential and Logarithmic Functions | 指数与对数函数
For a function like h(x) = eˣ / x, we use u = eˣ, v = x. Since u' = eˣ, v' = 1, we get:
对于 h(x) = eˣ / x 这样的函数，令 u = eˣ，v = x。由于 u' = eˣ，v' = 1，我们得到：
h '(x) = (eˣ·x − eˣ·1) / x² = eˣ(x − 1) / x²
When dealing with natural logarithms, consider k(x) = ln x / x. Here u = ln x, v = x, u' = 1/x, v' = 1. Then:
当遇到自然对数时，考虑 k(x) = ln x / x。此处 u = ln x，v = x，u' = 1/x，v' = 1。那么：
k '(x) = ((1/x)·x − ln x·1) / x² = (1 − ln x) / x²
These examples illustrate that the quotient rule works seamlessly with transcendental functions, and careful algebraic simplification is often required to reach the final form expected in IB exams.
这些例子表明，商法则与超越函数能够无缝衔接，而 IB 考试中通常需要仔细进行代数化简才能得到最终的答案形式。
8. Composite Functions and the Quotient Rule | 复合函数与商法则
ule. For instance, differentiate p(x) = sin(2x) / (x² + 1). The chain rule is needed for u' since u = sin(2x) gives u' = 2 cos(2x). The denominator v = x² + 1 has v' = 2x. Then:
当分子或分母本身包含复合函数时，必须将商法则与链式法则结合使用。例如，对 p(x) = sin(2x) / (x² + 1) 求导。由于 u = sin(2x)，链式法则给出 u' = 2 cos(2x)。分母 v = x² + 1，v' = 2x。于是：
p '(x) = (2 cos(2x)·(x²+1) − sin(2x)·2x) / (x²+1)²
Another common situation is a function like q(x) = √x / e3x. Here u = x½, so u' = (1/2)x−½; v = e3x requires the chain rule for v' = 3 e3x. Write the answer clearly, then simplify by factoring if possible.
另一种常见情况是 q(x) = √x / e3x 这类函数。其中 u = x½，u' = (1/2)x−½；v = e3x 需要链式法则得到 v' = 3 e3x。清晰地写出答案，然后尽可能通过提取公因式进行化简。
9. Finding Tangents and Normals | 求切线与法线
ific point. Suppose f(x) = x / (x² + 1) and you need the tangent at x = 1. First compute f '(x) using the quotient rule:
商法则在求特定点处的切线或法线方程问题中常被考查。假设 f(x) = x / (x² + 1)，需要求 x = 1 处的切线。先用商法则计算 f '(x)：
f '(x) = (1·(x²+1) − x·2x) / (x²+1)² = (1 − x²) / (x²+1)²
Evaluate at x = 1: f(1) = 1/2, f ' correctly apply the rule to obtain the gradient function before substituting the x-value.
在 x = 1 处求值：f(1) = 1/2，f '(1) = (1 − 1)/(2)² = 0。因此切线是水平的：y = 1/2，法线则是垂直的。这说明了在代入 x 值之前正确运用法则求出梯度函数有多么重要。
10. Common Mistakes | 常见错误
One frequent error is writing the numerator as u v' − v u' instead of v u' − u v'. Remembering the mnemonic "low d high" can help avoid this. Another mistake is forgetting to square the denominator; students sometimes write only v in the denominator. Additionally, when simplifying, do not cancel terms incorrectly across the subtraction; only factors can be cancelled, not terms. For example, (u' v − u v') / v² does not simplify to u' − u v' / v² without careful factoring.
一个常见错误是把分子写成 u v' − v u' 而不是 v u' − u v'。牢记“下乘上导”的口诀可以避免此类错误。另一个错误是忘记分母平方，有时学生只写了分母 v 而没有平方。此外，化简时切勿在减号两侧错误地随意消去；只有因式才能约分，项不可直接约去。例如，(u' v − u v') / v² 不能不经仔细提公因式就简化为 u' − u v' / v²。
It is also easy to mishandle the chain rule inside the quotient rule. Always differentiate the inner functions completely before substituting into the quotient formula. Taking time to write down u, u', v, v' separately will drastically reduce errors.
在商法则内部错误使用链式法则也很常见。务必先完整求导内层函数，再代入商法则公式。花点时间分别写出 u、u'、v、v' 会大幅减少错误。
11. Connection with the Product Rule | 与乘法法则的联系
ifferentiate r(x) = x / (2x+3). Write r(x) = x · (2x+3)⁻¹. Then, using the product rule: r ' Combining the fractions over a common denominator gives exactly the quotient rule result.
商法则与乘法法则有着深层的联系。由于 u/v = u · v⁻¹，只要你谨慎处理负指数，始终可以选择使用乘法法则。例如，对 r(x) = x / (2x+3) 求导，可写成 r(x) = x · (2x+3)⁻¹。然后用乘法法则：r '(x) = (1)·(2x+3)⁻¹ + x · (−1)(2x+3)⁻²·2 = 1/(2x+3) − 2x/(2x+3)²。通分合并分数后恰好得到商法则的结果。
This approach is particularly helpful when the denominator is a single simple power, but the quotient rule is more systematic for complicated denominators. Knowing both ways gives you flexibility during exams.
当分母仅为简单的幂函数时，这种方法尤其有用，但对于复杂分母，商法则更为系统化。掌握两种方法能在考试中为你提供灵活性。
12. Summary and Tips | 总结与学习建议
olving rational, trigonometric, exponential, and logarithmic functions. Always remember: f '(x) = (v u' − u v' ing expression as much as possible to reveal the neat form often required by mark schemes.
商法则是你在 IB 数学中会频繁使用的强大微分工具，特别是在涉及有理函数、三角函数、指数函数和对数函数的微积分问题中。请务必牢记：f '(x) = (v u' − u v') / v²。开始前先写下 u 和 v，仔细计算它们的导数，然后代入公式。尽可能化简所得表达式，以呈现评分方案通常要求的整洁形式。
Finally, practice with a variety of functions, including those requiring the chain rule inside the quotient rule. When you feel confident, try re-deriving the tangent of basic quotients to check your algebraic manipulation. With consistent practice, the quotient rule will become second nature.
最后，练习各种函数，包括那些需要在商法则内使用链式法则的类型。当你感到自信时，尝试重新推导基本商的切线，以检验自己的代数运算能力。通过持续练习，商法则将成为你的第二天性。
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