The Chain Rule · 链式法则
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| composite function/ˈkɒmpəzɪt ˈfʌŋkʃn/ | 复合函数 | fù hé hán shù |
| Chain Rule/tʃeɪn ruːl/ | 链式法则 | liàn shì fǎ zé |
| outer/ˈaʊtə/ | 外层 | wài céng |
| inner/ˈɪnə/ | 内层 | nèi céng |
Functions inside functions
- What is the derivative of $(3x+1)^{5}$? It's a power of a function — a composite function 复合函数.
- Expanding it would be brutal. There's a rule built exactly for this: the Chain Rule 链式法则.
- A composite is "an outer function wrapped around an inner one," like $f(g(x))$.
- The Chain Rule differentiates the layers one at a time and multiplies.
函数套函数
- $(3x+1)^{5}$ 的导数是什么?它是一个函数的幂——一个复合函数。
- 展开它会很痛苦。有一条正好为此而生的规则:链式法则。
- 复合函数是"一个外层函数裹着一个内层函数",如 $f(g(x))$。
- 链式法则一层一层地求导,再相乘。
The rule
- For $y=f(g(x))$:
-
$$\frac{d}{dx}\big[f(g(x))\big]=f'(g(x))\cdot g'(x)$$
- In words: derivative of the outer (leaving the inner alone) times the derivative of the inner.
- The little "$\cdot\, g'(x)$" is the piece people forget — it's the whole point.
规则
- 对 $y=f(g(x))$:
-
$$\frac{d}{dx}\big[f(g(x))\big]=f'(g(x))\cdot g'(x)$$
- 用文字说:外层的导数(内层不动)乘以内层的导数。
- 那个小小的"$\cdot\, g'(x)$"是人们会忘的部分——它才是关键。
A composite curve · 一个复合曲线
y = ax² + bx + c
A composite like $(3x+1)^2$ is steeper than its inner line alone — the chain rule multiplies the inner rate in. · 像 $(3x+1)^2$ 这样的复合函数比其内部直线本身更陡——链式法则将内部速率乘入。
The Chain Rule gives $\dfrac{d}{dx}[f(g(x))]=$ · 链式法则给出 $\dfrac{d}{dx}[f(g(x))]=$
Outer derivative at the inner, times the inner derivative. · 外层函数在内层处的导数，乘以内层函数的导数。
Spot the outer and inner
- Before differentiating, name the outer 外层 and inner 内层 functions.
- In $(3x+1)^5$: outer is "(something)$^5$", inner is $3x+1$.
- Outer derivative: $5(3x+1)^4$. Inner derivative: $3$.
- Multiply: $\dfrac{d}{dx}[(3x+1)^5]=5(3x+1)^4\cdot 3=15(3x+1)^4$.
认出外层与内层
- 求导前,先说出外层与内层函数。
- 在 $(3x+1)^5$ 中:外层是"(某物)$^5$",内层是 $3x+1$。
- 外层导数:$5(3x+1)^4$。内层导数:$3$。
- 相乘:$\dfrac{d}{dx}[(3x+1)^5]=5(3x+1)^4\cdot 3=15(3x+1)^4$。
For · 支持 $y=(2x+1)^3$, find $y'$ at $x=0$. (Chain rule: $3(2x+1)^2\cdot2$.) · 对于 $y=(2x+1)^3$，求 $y'$ 在 $x=0$ 处的值。（链式法则：$3(2x+1)^2\cdot2$。）
$y'=6(2x+1)^2$; at $0$: $6(1)=6$. · $y'=6(2x+1)^2$；在 $0$ 处：$6(1)=6$。
For · 支持 $\sqrt{x^2+1}=(x^2+1)^{1/2}$, select all · 所有 correct identifications. · 对于 $\sqrt{x^2+1}=(x^2+1)^{1/2}$，选择所有正确的识别项。
The square root is the outer; $x^2+1$ is the inner with derivative $2x$. · 平方根是外层；$x^2+1$ 是内层，其导数为 $2x$。
Chaining with the other rules
- The Chain Rule stacks with everything: power, product, quotient, and elementary derivatives.
- $\dfrac{d}{dx}[\sin(x^2)]=\cos(x^2)\cdot 2x$ — outer $\sin$, inner $x^2$.
- $\dfrac{d}{dx}[e^{3x}]=e^{3x}\cdot 3=3e^{3x}$.
- For deeply nested functions, peel one layer at a time, multiplying each inner derivative as you go.
与其他规则串联
- 链式法则与一切叠加:幂、乘积、商、基本导数。
- $\dfrac{d}{dx}[\sin(x^2)]=\cos(x^2)\cdot 2x$——外层 $\sin$,内层 $x^2$。
- $\dfrac{d}{dx}[e^{3x}]=e^{3x}\cdot 3=3e^{3x}$。
- 对深度嵌套的函数,一次剥一层,边剥边乘上每个内层导数。
What is $\dfrac{d}{dx}[\sin(x^2)]$? · $\dfrac{d}{dx}[\sin(x^2)]$是什么？
Outer $\cos(x^2)$ times inner derivative $2x$. · 外层 $\cos(x^2)$ 乘以内层导数 $2x$。
The factor most often forgotten in the chain rule is the ____ derivative $g'(x)$. · 链式法则中最常被遗忘的因素是____导数 $g'(x)$。
You must multiply by $g'(x)$ after differentiating the outer. · 对外层求导后必须乘以 $g'(x)$。
What is $\dfrac{d}{dx}[e^{3x}]$? · $\dfrac{d}{dx}[e^{3x}]$是什么？
Outer $e^{3x}$ times inner derivative $3$. · 外层 $e^{3x}$ 乘以内层导数 $3$。
Never forget to multiply by the inner derivative $g'(x)$. $\frac{d}{dx}[\sin(x^2)]$ is $\cos(x^2)\cdot 2x$, not just $\cos(x^2)$. Leaving off the "$\cdot\,g'(x)$" is the single most common calculus error — the outer derivative alone is only half the answer.
千万别忘了乘以内层导数 $g'(x)$。$\frac{d}{dx}[\sin(x^2)]$ 是 $\cos(x^2)\cdot 2x$,而不是只有 $\cos(x^2)$。漏掉"$\cdot\,g'(x)$"是微积分里最常见的错误——只有外层导数只是一半答案。
Differentiate $y=\sqrt{x^2+1}$.
- Rewrite: $y=(x^2+1)^{1/2}$. Outer: $(\ )^{1/2}$; inner: $x^2+1$.
- Outer derivative: $\tfrac12(x^2+1)^{-1/2}$. Inner derivative: $2x$.
- $y'=\tfrac12(x^2+1)^{-1/2}\cdot 2x=\dfrac{x}{\sqrt{x^2+1}}$.
对 $y=\sqrt{x^2+1}$ 求导。
- 改写:$y=(x^2+1)^{1/2}$。外层:$(\ )^{1/2}$;内层:$x^2+1$。
- 外层导数:$\tfrac12(x^2+1)^{-1/2}$。内层导数:$2x$。
- $y'=\tfrac12(x^2+1)^{-1/2}\cdot 2x=\dfrac{x}{\sqrt{x^2+1}}$。
The Chain Rule differentiates a composite function: $\frac{d}{dx}[f(g(x))]=f'(g(x))\cdot g'(x)$ — derivative of the outer (inner untouched) times the derivative of the inner. Identify the layers first, and never drop the $\cdot\,g'(x)$ factor. It combines with every other rule for nested expressions.
链式法则对复合函数求导:$\frac{d}{dx}[f(g(x))]=f'(g(x))\cdot g'(x)$——外层的导数(内层不动)乘以内层的导数。先认出各层,永远别丢掉 $\cdot\,g'(x)$ 这个因子。它对嵌套表达式与其他每条规则组合。