Selecting Procedures for Calculating Derivatives · 选择计算导数的方法
You have a toolbox — now choose the tool
- By now you own power, constant-multiple, sum/difference, product, quotient, chain, and elementary derivative rules.
- The exam skill is picking the right rule fast, and combining rules in the correct order.
- A messy-looking derivative is usually easy once you see its structure.
- This lesson is a decision guide for reading that structure.
你有一个工具箱——现在选对工具
- 到现在你已拥有幂、常数倍、和/差、乘积、商、链式和基本导数规则。
- 考试的技巧是快速选对规则,并按正确顺序组合规则。
- 一个看起来杂乱的导数,一旦看清它的结构,通常很容易。
- 这一课是读懂那个结构的决策指南。
Read the outermost structure first
- Ask: at the very top level, is this a sum, a product, a quotient, or a composition?
- Sum/difference → differentiate term by term.
- Product → product rule; quotient → quotient rule.
- Composition (function inside function) → chain rule, peeling outer to inner.
先读最外层结构
- 问:在最顶层,这是一个和、一个乘积、一个商,还是一个复合?
- 和/差 → 逐项求导。
- 乘积 → 乘积法则;商 → 商法则。
- 复合(函数套函数)→ 链式法则,从外向内剥。
Read the structure, pick the rule · 阅读结构，选择规则
y = a / (x − b)
Whether a curve comes from a product, quotient, or composition changes which rule differentiates it — read the top level first. · 曲线源自乘积、商还是复合运算决定了使用哪个规则进行求导——首先阅读顶层结构。
Which rule does $y=x^2\sin x$ need at the top level? · $y=x^2\sin x$ 在顶层需要哪种规则？
It is a product of $x^2$ and $\sin x$ → product rule. · 它是 $x^2$ 与 $\sin x$ 的乘积 → 乘积法则。
The top-level structure of $\dfrac{\sin x}{x^2}$ is a... · $\dfrac{\sin x}{x^2}$ 的顶层结构是一个...
It is · 它是 $\sin x$ divided by $x^2$ — a quotient (or rewrite as $x^{-2}\sin x$). · 它是 $\sin x$ 除以 $x^2$ —— 一个商（或重写为 $x^{-2}\sin x$）。
Match each function to the top-level rule it needs. · 将每个函数与其所需的顶层规则匹配。
Read the outermost operation: term-by-term, product, quotient, or composition. · 阅读最外层运算：逐项、乘积、商或复合。
Combine rules in the right order
- Real functions nest rules: a product whose factors are composites, a quotient with a chain inside.
- Work outside-in: apply the top-level rule first, then handle each sub-piece with its own rule.
- $\dfrac{d}{dx}\big[x^2\sin(3x)\big]$: product rule first, and the $\sin(3x)$ factor needs the chain rule → $2x\sin(3x)+x^2\cos(3x)\cdot3$.
- Keep sub-derivatives in labeled boxes so nothing gets dropped.
按正确顺序组合规则
- 真实函数会嵌套规则:因子是复合函数的乘积、内部含链式的商。
- 从外向内做:先套用顶层规则,再用各自的规则处理每个子块。
- $\dfrac{d}{dx}\big[x^2\sin(3x)\big]$:先用乘积法则,而 $\sin(3x)$ 因子需要链式法则 → $2x\sin(3x)+x^2\cos(3x)\cdot3$。
- 把子导数放在标注的"盒子"里,免得漏掉任何一个。
Simplify $\dfrac{x^2+x}{x}$ first, then differentiate. What is the derivative? · 先简化 $\dfrac{x^2+x}{x}$，再求导。导数是什么？
$\frac{x^2+x}{x}=x+1$, so the derivative is $1$ — no quotient rule needed. · $\frac{x^2+x}{x}=x+1$，因此导数为 $1$ —— 不需要商法则。
For nested functions, apply rules working from the ____ level inward. · 对于嵌套函数，从____层向内应用规则。
Handle the outermost structure first, then each sub-piece. · 先处理最外层结构，然后逐个处理每个子部分。
Simplify smart, then sanity-check
- Before diving in, ask if algebra makes it easier: expand a small product, or split a fraction into terms.
- $\dfrac{x^2+x}{x}=x+1$, so its derivative is just $1$ — no quotient rule needed.
- After differentiating, check for reasonableness: right number of terms, signs sensible, units plausible.
- Choosing well up front beats grinding through the hardest rule by reflex.
聪明地化简,再检验
- 动手前先问代数能否让它更简单:展开一个小乘积,或把分式拆成几项。
- $\dfrac{x^2+x}{x}=x+1$,所以它的导数就是 $1$——不需要商法则。
- 求导后,检查合理性:项数对不对、符号是否说得通、单位是否合理。
- 一开始就选对,胜过条件反射地硬套最难的规则。
After differentiating, checking the answer for reasonableness (term count, signs) is good practice. · 求导后，检查答案的合理性（项数、符号）是好习惯。
A quick sanity check catches many slips. · 快速 sanity check 可以发现许多错误。
Match the rule to the top-level structure, not to the first symbol you see. $\dfrac{\sin x}{x^2}$ is a quotient (use the quotient rule or rewrite as $x^{-2}\sin x$ for the product rule) — it is not just "the derivative of $\sin$." Misreading the structure sends you down the wrong rule entirely.
把规则匹配到顶层结构,而不是你看到的第一个符号。$\dfrac{\sin x}{x^2}$ 是一个商(用商法则,或改写成 $x^{-2}\sin x$ 用乘积法则)——它不是只是"$\sin$ 的导数"。读错结构会把你引向完全错误的规则。
Differentiate $y=\dfrac{e^{x}}{x^2+1}$.
- Top level is a quotient: $f=e^x$, $g=x^2+1$.
- $f'=e^x$, $g'=2x$ (the $g'$ is a simple power-rule sub-step).
- $y'=\dfrac{e^x(x^2+1)-e^x(2x)}{(x^2+1)^2}=\dfrac{e^x(x^2-2x+1)}{(x^2+1)^2}$.
对 $y=\dfrac{e^{x}}{x^2+1}$ 求导。
- 顶层是商:$f=e^x$,$g=x^2+1$。
- $f'=e^x$,$g'=2x$($g'$ 是简单的幂法则子步骤)。
- $y'=\dfrac{e^x(x^2+1)-e^x(2x)}{(x^2+1)^2}=\dfrac{e^x(x^2-2x+1)}{(x^2+1)^2}$。
Selecting a procedure: read the top-level structure — sum, product, quotient, or composition — and apply that rule first, then handle each sub-piece with its own rule, working outside-in. Simplify with algebra when it's cheaper, and check the result for reasonableness.
选择步骤:读顶层结构——和、乘积、商或复合——先套用那条规则,再用各自的规则处理每个子块,从外向内做。当代数更省事时就先化简,并检查结果的合理性。