Derivative Rules: Constant, Sum, Difference, and Constant Multiple · 求导法则：常数法则、和差法则及常数倍法则
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| polynomial/ˌpɒlɪˈnəʊmɪəl/ | 多项式 | duō xiàng shì |
Four rules that break a polynomial apart
- The Power Rule handles one term. To differentiate a whole polynomial 多项式 you need to combine terms.
- Four simple rules let you do it term by term, in your head.
- They cover constants, sums, differences, and coefficients.
- Together with the Power Rule, they differentiate any polynomial in one pass.
四条把多项式拆开的规则
- 幂法则处理一项。要对整个多项式求导,你需要把各项组合起来。
- 四条简单规则让你逐项处理,在脑中就能完成。
- 它们涵盖常数、和、差与系数。
- 与幂法则一起,一遍就能对任何多项式求导。
Constant, sum, and difference
- Constant rule: the derivative of a constant is $0$. A flat line has zero slope: $\dfrac{d}{dx}[7]=0$.
- Sum rule: $\dfrac{d}{dx}[f+g]=f'+g'$ — differentiate each piece and add.
- Difference rule: $\dfrac{d}{dx}[f-g]=f'-g'$ — same, with a minus.
- So you can split a long expression and handle each term separately.
常数、和、差
- 常数法则: 常数的导数是 $0$。水平线斜率为零:$\dfrac{d}{dx}[7]=0$。
- 和法则: $\dfrac{d}{dx}[f+g]=f'+g'$——对每部分求导再相加。
- 差法则: $\dfrac{d}{dx}[f-g]=f'-g'$——一样,只是减号。
- 所以你可以把长表达式拆开,逐项处理。
What is $\dfrac{d}{dx}[12]$? · $\dfrac{d}{dx}[12]$是什么？
The derivative of any constant is $0$. · 任何常数的导数是 $0$。
By the sum rule, $\dfrac{d}{dx}[f+g]=f'+$ ____. · 根据和法则，$\dfrac{d}{dx}[f+g]=f'+$ ____。
Differentiate each piece and add. · 分别对每一部分求导然后相加。
Constant multiple
- Constant multiple rule: a coefficient just rides along: $\dfrac{d}{dx}[k\,f]=k\,f'$.
- $\dfrac{d}{dx}[5x^3]=5\cdot 3x^2=15x^2$.
- The number out front is untouched by differentiation; only the $x$-part changes.
- Combine with the sum rule: $\dfrac{d}{dx}[5x^3-4x]=15x^2-4$.
常数倍
- 常数倍法则: 系数只是"搭便车":$\dfrac{d}{dx}[k\,f]=k\,f'$。
- $\dfrac{d}{dx}[5x^3]=5\cdot 3x^2=15x^2$。
- 前面的数字在求导时不变;只有 $x$ 的部分变化。
- 与和法则结合:$\dfrac{d}{dx}[5x^3-4x]=15x^2-4$。
A cubic and its changing slope · 一个三次函数及其变化的斜率
y = ax³ + bx
The derivative of a polynomial is another polynomial — drag the coefficients and picture where the slope is positive or negative. · 多项式的导数仍是多项式——拖动系数并想象斜率为正或负的位置。
If $f(x)=5x^3$, find $f'(2)$. · 若 $f(x)=5x^3$，求 $f'(2)$。
$f'(x)=15x^2$, so $f'(2)=15\cdot4=60$. · $f'(x)=15x^2$，所以 $f'(2)=15\cdot4=60$。
For · 支持 $\dfrac{d}{dx}[4x^2+3x-8]$, select all · 所有 correct term derivatives. · 对于 $\dfrac{d}{dx}[4x^2+3x-8]$，选择所有正确的项导数。
The constant $-8$ differentiates to $0$, not $-8$. · 常数 $-8$ 的导数是 $0$，而不是 $-8$。
Differentiate a polynomial in one pass
- Read the polynomial term by term, applying the Power Rule and coefficients as you go.
- $\dfrac{d}{dx}[3x^4-2x^2+7x-9]=12x^3-4x+7-0=12x^3-4x+7$.
- Notice the constant $-9$ vanishes, and the $7x$ becomes just $7$.
- With practice you write the derivative directly, no scratch work.
一遍对多项式求导
- 逐项读多项式,边读边套用幂法则和系数。
- $\dfrac{d}{dx}[3x^4-2x^2+7x-9]=12x^3-4x+7-0=12x^3-4x+7$。
- 注意常数 $-9$ 消失,$7x$ 变成 $7$。
- 熟练之后你直接写出导数,不用打草稿。
Differentiate $f(x)=3x^4-2x^2+7x-9$. · 对 $f(x)=3x^4-2x^2+7x-9$ 求导。
Term by term: $12x^3$, $-4x$, $+7$, and the constant $\to 0$. · 逐项求导：$12x^3$，$-4x$，$+7$，以及常数项 $\to 0$。
The sum rule lets you compute $\dfrac{d}{dx}[f\cdot g]$ as $f'\cdot g'$. · 和法则允许你将 $\dfrac{d}{dx}[f\cdot g]$ 计算为 $f'\cdot g'$。
Sum/difference rules split across $+$ and $-$ only; a product needs the Product Rule. · 和/差法则仅适用于拆分 $+$ 和 $-$；乘积需要使用乘积法则。
There is no "product rule shortcut" hiding here: $\frac{d}{dx}[f\cdot g]$ is not $f'\cdot g'$. The sum/difference rules split across $+$ and $-$ only. Products and quotients need their own rules (lessons 2.8–2.9). Also, the derivative of a constant is $0$, not the constant itself.
这里没有藏着什么"乘积法则捷径":$\frac{d}{dx}[f\cdot g]$ 不是 $f'\cdot g'$。和/差法则只在 $+$ 与 $-$ 上拆分。乘积和商需要它们各自的规则(2.8–2.9 课)。此外,常数的导数是 $0$,而不是常数本身。
Differentiate $p(x)=6x^3+\dfrac{4}{x^2}-x+10$.
- Rewrite: $p(x)=6x^3+4x^{-2}-x+10$.
- Term by term: $6\cdot3x^2$, $4\cdot(-2)x^{-3}$, $-1$, and $0$.
- $p'(x)=18x^2-8x^{-3}-1=18x^2-\dfrac{8}{x^3}-1$.
对 $p(x)=6x^3+\dfrac{4}{x^2}-x+10$ 求导。
- 改写:$p(x)=6x^3+4x^{-2}-x+10$。
- 逐项:$6\cdot3x^2$、$4\cdot(-2)x^{-3}$、$-1$、$0$。
- $p'(x)=18x^2-8x^{-3}-1=18x^2-\dfrac{8}{x^3}-1$。
Four combining rules: constant $\to 0$; sum/difference differentiate term by term; constant multiple keeps the coefficient out front ($\frac{d}{dx}[kf]=kf'$). With the Power Rule these differentiate any polynomial in a single pass — but they split only across $+$ and $-$, never across a product.
四条组合规则:常数 $\to 0$;和/差逐项求导;常数倍把系数留在前面($\frac{d}{dx}[kf]=kf'$)。与幂法则一起,一遍就能对任何多项式求导——但它们只在 $+$ 与 $-$ 上拆分,绝不在乘积上。