The Quotient Rule · 商法则
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| Quotient Rule/ˈkwəʊʃənt ruːl/ | 商法则 | shāng fǎ zé |
| rational expression/ˈræʃənl ekˈspreʃn/ | 有理式 | yǒu lǐ shì |
Dividing functions needs care
- Just as products need the Product Rule, quotients need the Quotient Rule 商法则.
- You can't differentiate top and bottom separately and divide — that's wrong.
- The rule handles a rational expression 有理式 $\dfrac{f}{g}$ of two functions.
- Its numerator has a subtraction, so order and signs matter a lot.
函数相除要小心
- 正如乘积需要乘积法则,商需要商法则。
- 你不能分别对上下求导再相除——那是错的。
- 这条规则处理两个函数的有理式 $\dfrac{f}{g}$。
- 它的分子有一个减法,所以顺序和符号非常重要。
The rule
-
$$\frac{d}{dx}\!\left[\frac{f}{g}\right]=\frac{f'\,g-f\,g'}{g^{2}}$$
- Top: "derivative of the top times the bottom, minus the top times derivative of the bottom."
- Bottom: the original denominator, squared.
- A memory hook: "low d-high minus high d-low, over low-squared."
规则
-
$$\frac{d}{dx}\!\left[\frac{f}{g}\right]=\frac{f'\,g-f\,g'}{g^{2}}$$
- 分子:"上的导数乘下,减去上乘下的导数。"
- 分母:原来的分母,平方。
- 一个记忆口诀:"下乘上导 减 上乘下导,除以下的平方。"
The Quotient Rule numerator for $\dfrac{f}{g}$ is... · $\dfrac{f}{g}$ 的商法则分子是……
Numerator is $f'g-fg'$, over $g^2$. · 分子是 $f'g-fg'$，除以 $g^2$。
In the Quotient Rule, the denominator of the answer is $g$ ____ (to what power?). · 在商法则中，答案的分母是 $g$ ____（到几次方？）
The denominator is $g^2$. · 分母是 $g^2$。
Select all · 所有 true statements about the Quotient Rule. · 选择关于商法则的所有正确陈述。
$g^2\ge0$ always, so it never flips the sign. · $g^2\ge0$ 始终为正，因此永远不会翻转符号。
Order is everything
- The numerator is $f'g - fg'$, not $fg' - f'g$ — swapping them flips every sign.
- Always write the $f'g$ term first; that fixes the order and prevents sign slips.
- The denominator $g^2$ is always positive, so it never changes the sign of the answer.
- Keep the pieces labeled: $f,f',g,g'$ before you assemble.
顺序至关重要
- 分子是 $f'g - fg'$,不是 $fg' - f'g$——交换它们会翻转每个符号。
- 永远先写 $f'g$ 那一项;这就固定了顺序,防止符号出错。
- 分母 $g^2$ 永远为正,所以它绝不会改变答案的符号。
- 组装前先给各部分标号:$f,f',g,g'$。
A rational curve to differentiate · 一个需要求导的有理曲线
y = a / (x − b)
A quotient like $\dfrac{x^2}{x+1}$ has a slope the Quotient Rule computes — note the vertical asymptote where the denominator is zero. · 像 $\dfrac{x^2}{x+1}$ 这样的商由商法则计算斜率——注意分母为零处的垂直渐近线。
Writing the numerator as $fg' - f'g$ (reversed) gives the same answer. · 将分子写成 $fg' - f'g$（颠倒顺序）得到相同的答案。
Reversing the order negates the whole numerator — a sign error. · 颠倒顺序会取反整个分子——这是一个符号错误。
Differentiate $y=\dfrac{x}{x+1}$. · 对 $y=\dfrac{x}{x+1}$ 求导。
$\frac{(1)(x+1)-x(1)}{(x+1)^2}=\frac{1}{(x+1)^2}$.
Sometimes a product is easier
- A quotient $\dfrac{f}{g}$ can be rewritten as a product $f\cdot g^{-1}$ and done with the Product + Power rules.
- For simple denominators this can be quicker: $\dfrac{x}{x^2}=x^{-1}\Rightarrow -x^{-2}$.
- But for genuine two-function quotients like $\dfrac{\sin x}{x}$, the Quotient Rule is the clean path.
- Choose whichever avoids the messier algebra.
有时改成乘积更容易
- 商 $\dfrac{f}{g}$ 可改写成乘积 $f\cdot g^{-1}$,用乘积法则 + 幂法则来做。
- 对简单分母,这可能更快:$\dfrac{x}{x^2}=x^{-1}\Rightarrow -x^{-2}$。
- 但对真正的两函数商(如 $\dfrac{\sin x}{x}$),商法则才是干净的路。
- 选能避开更麻烦代数的那个。
The quotient $\dfrac{x}{x^2}$ is fastest differentiated by... · 最快求导商 $\dfrac{x}{x^2}$ 的方法是……
$\frac{x}{x^2}=x^{-1}\Rightarrow -x^{-2}$ — simpler than the Quotient Rule here. · $\frac{x}{x^2}=x^{-1}\Rightarrow -x^{-2}$——比此处使用商法则更简单。
The Quotient Rule numerator is $f'g - fg'$ — the order and the minus sign are the whole difficulty. Writing $fg' - f'g$ (reversed) negates your answer. And do not forget to square the denominator. Both mistakes are extremely common under time pressure.
商法则的分子是 $f'g - fg'$——顺序和减号就是全部难点。写成 $fg' - f'g$(反过来)会让答案变号。而且别忘了把分母平方。这两个错误在时间压力下极其常见。
Differentiate $y=\dfrac{x^2}{x+1}$.
- $f=x^2\Rightarrow f'=2x$; $\quad g=x+1\Rightarrow g'=1$.
- $y'=\dfrac{f'g-fg'}{g^2}=\dfrac{2x(x+1)-x^2(1)}{(x+1)^2}$.
- $=\dfrac{2x^2+2x-x^2}{(x+1)^2}=\dfrac{x^2+2x}{(x+1)^2}$.
对 $y=\dfrac{x^2}{x+1}$ 求导。
- $f=x^2\Rightarrow f'=2x$;$\quad g=x+1\Rightarrow g'=1$。
- $y'=\dfrac{f'g-fg'}{g^2}=\dfrac{2x(x+1)-x^2(1)}{(x+1)^2}$。
- $=\dfrac{2x^2+2x-x^2}{(x+1)^2}=\dfrac{x^2+2x}{(x+1)^2}$。
The Quotient Rule: $\frac{d}{dx}\!\left[\frac{f}{g}\right]=\frac{f'g-fg'}{g^{2}}$ — "low d-high minus high d-low, over low-squared." Keep the numerator's order ($f'g$ first) and square the denominator. For simple cases, rewriting as a product $f g^{-1}$ can be easier.
商法则:$\frac{d}{dx}\!\left[\frac{f}{g}\right]=\frac{f'g-fg'}{g^{2}}$——"下乘上导减上乘下导,除以下的平方"。保持分子的顺序($f'g$ 在前)并把分母平方。对简单情形,改写成乘积 $f g^{-1}$ 可能更容易。