Derivatives of cos x, sin x, eˣ, and ln x · cos x, sin x, eˣ 和 ln x 的导数
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| exponential/ˌekspəˈnenʃl/ | 指数函数 | zhǐ shù hán shù |
| logarithmic/ˌlɒɡəˈrɪθmɪk/ | 对数 | duì shù |
| elementary derivatives/ˌelɪˈmentəri dɪˈrɪvətɪvz/ | 基本导数 | jī běn dǎo shù |
Four derivatives worth memorizing
- The Power Rule stops at powers of $x$. Beyond it lie the elementary functions.
- Four of them appear constantly, so their derivatives are worth knowing cold.
- They are $\sin x$, $\cos x$, $e^x$, and $\ln x$.
- Learn these four and, with the earlier rules, you can differentiate a huge range of expressions.
四个值得背下来的导数
- 幂法则止步于 $x$ 的幂。它之外是基本函数。
- 其中四个反复出现,所以它们的导数值得烂熟于心。
- 它们是 $\sin x$、$\cos x$、$e^x$、$\ln x$。
- 记住这四个,再配合前面的规则,你就能对海量表达式求导。
The trig pair
- $\dfrac{d}{dx}[\sin x]=\cos x$.
- $\dfrac{d}{dx}[\cos x]=-\sin x$ — note the minus sign.
- A handy check: the slope of $\sin x$ at $0$ is $1$ (steepest rise), and $\cos 0=1$. ✓
- (These require $x$ in radians — the clean formulas only hold there.)
三角函数一对
- $\dfrac{d}{dx}[\sin x]=\cos x$。
- $\dfrac{d}{dx}[\cos x]=-\sin x$——注意那个负号。
- 一个方便的检验:$\sin x$ 在 $0$ 处的斜率是 $1$(上升最陡),而 $\cos 0=1$。✓
- (这些要求 $x$ 用弧度——干净的公式只在弧度下成立。)
The slope of sine is cosine · 正弦的斜率是余弦
Where · 何地 $\sin x$ is steepest (at $0$) its slope is $1$ — exactly $\cos 0$. The derivative of $\sin$ is $\cos$. · 当 $\sin x$ 最陡时（在 $0$ 处），其斜率为 $1$——恰好等于 $\cos 0$。$\sin$ 的导数是 $\cos$。
What is $\dfrac{d}{dx}[\sin x]$? · $\dfrac{d}{dx}[\sin x]$是什么？
$\frac{d}{dx}[\sin x]=\cos x$.
$\dfrac{d}{dx}[\cos x]=$ ____ $\sin x$ (mind the sign). · $\dfrac{d}{dx}[\cos x]=$ ____ $\sin x$（注意符号）。
The derivative of cosine is $-\sin x$. · 余弦的导数是 $-\sin x$。
The exponential and the logarithm
- $\dfrac{d}{dx}[e^x]=e^x$ — the exponential 指数函数 function is its own derivative. Its slope always equals its height.
- $\dfrac{d}{dx}[\ln x]=\dfrac{1}{x}$ — the logarithmic 对数 derivative, valid for $x>0$.
- $e^x$ is the unique function that never changes under differentiation.
- $\ln x$ turns into the simple reciprocal $\tfrac1x$ — a surprisingly tidy result.
指数与对数
- $\dfrac{d}{dx}[e^x]=e^x$——指数函数是它自己的导数。它的斜率永远等于它的高度。
- $\dfrac{d}{dx}[\ln x]=\dfrac{1}{x}$——对数导数,对 $x>0$ 有效。
- $e^x$ 是求导下唯一不变的函数。
- $\ln x$ 变成简单的倒数 $\tfrac1x$——一个出奇整洁的结果。
$e^x$ is its own derivative. · $e^x$ 是其自身的导数。
$\frac{d}{dx}[e^x]=e^x$ — unique among functions. · $\frac{d}{dx}[e^x]=e^x$——在函数中独一无二。
What is $\dfrac{d}{dx}[\ln x]$ (for $x>0$)? · $\dfrac{d}{dx}[\ln x]$ (针对 $x>0$) 是什么？
$\frac{d}{dx}[\ln x]=\tfrac1x$.
Mix them with the combining rules
- These elementary derivatives 基本导数 slot straight into the sum, difference, and constant-multiple rules.
- $\dfrac{d}{dx}[3\sin x-2e^x]=3\cos x-2e^x$.
- $\dfrac{d}{dx}[x^2+\ln x]=2x+\dfrac1x$.
- Differentiate each term with its own rule, then combine — nothing new to learn beyond the four facts.
与组合规则混用
- 这些基本导数可直接嵌入和、差、常数倍法则。
- $\dfrac{d}{dx}[3\sin x-2e^x]=3\cos x-2e^x$。
- $\dfrac{d}{dx}[x^2+\ln x]=2x+\dfrac1x$。
- 用各自的规则对每项求导,再组合——除这四个事实外,没有新东西要学。
Differentiate $3\sin x-2e^x$. · 对 $3\sin x-2e^x$ 求导。
$3\cos x$ and $e^x$ stays $e^x$: $3\cos x-2e^x$. · $3\cos x$ 和 $e^x$ 保持不变：$e^x$：$3\cos x-2e^x$。
Select all · 所有 correct elementary derivatives. · 选择所有正确的初等函数导数。
The last is wrong — $\frac{d}{dx}[\ln x]=\tfrac1x$, not $x$. · 最后一项是错误的——应为 $\frac{d}{dx}[\ln x]=\tfrac1x$，而非 $x$。
Mind the signs and the domain. $\frac{d}{dx}[\cos x]=-\sin x$ (the minus is the top mistake). $e^x$ is its own derivative — do not apply the Power Rule to it. And $\frac{d}{dx}[\ln x]=\tfrac1x$ holds only for $x>0$, the domain of $\ln$.
注意符号和定义域。$\frac{d}{dx}[\cos x]=-\sin x$(那个负号是头号错误)。$e^x$ 是它自己的导数——别对它用幂法则。而 $\frac{d}{dx}[\ln x]=\tfrac1x$ 只在 $x>0$($\ln$ 的定义域)成立。
Differentiate $h(x)=4\cos x+e^x-5\ln x$.
- $\dfrac{d}{dx}[4\cos x]=4(-\sin x)=-4\sin x$.
- $\dfrac{d}{dx}[e^x]=e^x$; $\quad\dfrac{d}{dx}[-5\ln x]=-\dfrac{5}{x}$.
- $h'(x)=-4\sin x+e^x-\dfrac{5}{x}$.
对 $h(x)=4\cos x+e^x-5\ln x$ 求导。
- $\dfrac{d}{dx}[4\cos x]=4(-\sin x)=-4\sin x$。
- $\dfrac{d}{dx}[e^x]=e^x$;$\quad\dfrac{d}{dx}[-5\ln x]=-\dfrac{5}{x}$。
- $h'(x)=-4\sin x+e^x-\dfrac{5}{x}$。
Memorize four elementary derivatives: $\frac{d}{dx}[\sin x]=\cos x$, $\frac{d}{dx}[\cos x]=-\sin x$, $\frac{d}{dx}[e^x]=e^x$, $\frac{d}{dx}[\ln x]=\tfrac1x$. They combine with the sum/difference/constant-multiple rules to differentiate mixed expressions. Watch the minus on $\cos$, keep $x$ in radians, and remember $e^x$ is its own derivative.
背下四个基本导数:$\frac{d}{dx}[\sin x]=\cos x$、$\frac{d}{dx}[\cos x]=-\sin x$、$\frac{d}{dx}[e^x]=e^x$、$\frac{d}{dx}[\ln x]=\tfrac1x$。它们与和/差/常数倍法则组合,对混合表达式求导。注意 $\cos$ 的负号,$x$ 保持弧度,并记住 $e^x$ 是它自己的导数。