Defining the Derivative of a Function and Using Derivative Notation · 定义函数的导数并使用导数记号
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| derivative/dɪˈrɪvətɪv/ | 导数 | dǎo shù |
The derivative: instantaneous rate, packaged
- The instantaneous rate of change is so useful it gets its own name: the derivative 导数.
- It turns "slope of the tangent at each $x$" into a brand-new function, $f'(x)$.
- Feed it an $x$, and it returns the slope of $f$ right there.
- This lesson makes the definition precise and shows the notations you'll use everywhere.
导数:把瞬时变化率打包
- 瞬时变化率如此有用,以至于它有了自己的名字:导数。
- 它把"每个 $x$ 处切线的斜率"变成一个全新的函数 $f'(x)$。
- 给它一个 $x$,它就返回 $f$ 在那里的斜率。
- 这一课把定义讲精确,并展示你处处都会用到的记号。
The derivative is the tangent slope · 导数是切线斜率
y = x²
Drag the point along the curve — the number the tangent slope shows is exactly $f'(x)$ at that input. · 拖动点沿曲线移动——切线斜率显示的数字正好是该输入处的$f'(x)$。
The limit definition
- The derivative of $f$ is
-
$$f'(x)=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}$$
- Inside is a difference quotient — the secant slope over an interval of width $h$.
- Taking $h\to0$ shrinks the interval to nothing, turning the secant slope into the tangent slope.
极限定义
- $f$ 的导数是
-
$$f'(x)=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}$$
- 里面是一个差商——宽度为 $h$ 的区间上的割线斜率。
- 取 $h\to0$ 把区间缩到零,把割线斜率变成切线斜率。
Which expression is the limit definition of $f'(x)$? · 哪个表达式是$f'(x)$的极限定义？
The derivative is the limit of the difference quotient as $h\to0$. · 导数是差商在$h\to0$时的极限。
The "at a point" version
- To find the slope at one specific input $a$, use the alternate form:
-
$$f'(a)=\lim_{x\to a}\frac{f(x)-f(a)}{x-a}$$
- Here $x$ slides toward $a$ instead of a step $h$ shrinking to $0$ — same idea, different bookkeeping.
- Both definitions are just "secant slope, in the limit," so they always agree.
"在某点"的版本
- 要求某个特定输入 $a$ 处的斜率,用替代形式:
-
$$f'(a)=\lim_{x\to a}\frac{f(x)-f(a)}{x-a}$$
- 这里是 $x$ 滑向 $a$,而不是步长 $h$ 缩向 $0$——同一想法,不同记法。
- 两个定义都是"极限意义下的割线斜率",所以它们总是一致。
You can compute $f'(x)$ by setting $h=0$ directly in $\dfrac{f(x+h)-f(x)}{h}$. · 您可以通过在 $f'(x)$ 中直接设置 $h=0$ 来计算 $\dfrac{f(x+h)-f(x)}{h}$.
That gives $\tfrac00$; you must take the limit after simplifying. · 这会给出$\tfrac00$；你必须先化简再取极限。
The alternate form $f'(a)=\lim_{x\to a}\dfrac{f(x)-f(a)}{x-a}$ is best described as... · 替代形式$f'(a)=\lim_{x\to a}\dfrac{f(x)-f(a)}{x-a}$最恰当的描述是...
It is the same tangent slope at $a$, expressed with $x\to a$ instead of $h\to0$. · 这是在$a$处的同一切线斜率，用$x\to a$代替$h\to0$表达。
Reading the notations
- The same derivative wears several outfits — learn to recognize all of them:
- Lagrange: $f'(x)$ (read "$f$ prime of $x$").
- Leibniz: $\dfrac{dy}{dx}$ or $\dfrac{d}{dx}[f(x)]$ — emphasizes "rate of $y$ with respect to $x$."
- All mean the instantaneous rate of change, i.e. the slope of the tangent line at $x$.
认识各种记号
- 同一个导数穿着好几套"外衣"——要学会认出它们全部:
- 拉格朗日记号: $f'(x)$(读作"$f$ 撇 $x$")。
- 莱布尼茨记号: $\dfrac{dy}{dx}$ 或 $\dfrac{d}{dx}[f(x)]$——强调"$y$ 关于 $x$ 的变化率"。
- 它们都表示瞬时变化率,即 $x$ 处切线的斜率。
The Leibniz notation $\dfrac{dy}{dx}$ means the same thing as $f'(x)$: the ____ of the tangent line. · 莱布尼茨记号$\dfrac{dy}{dx}$与$f'(x)$含义相同：即切线的____。
Both denote the instantaneous rate of change = tangent slope. · 两者都表示瞬时变化率 = 切线斜率。
Select all · 所有 notations that denote the derivative of $y=f(x)$. · 选择所有表示$y=f(x)$导数的记号。
The first three are all the derivative; $f(x)^{-1}$ is a reciprocal/inverse, not a derivative. · 前三个都是导数；$f(x)^{-1}$是倒数/逆运算，而非导数。
You cannot just plug $h=0$ into $\frac{f(x+h)-f(x)}{h}$ — that gives $\tfrac00$. The derivative is the limit as $h\to0$, which you evaluate by simplifying the difference quotient first (cancel the $h$), then letting $h\to0$. Skipping the limit is the classic mistake.
你不能直接把 $h=0$ 代入 $\frac{f(x+h)-f(x)}{h}$——那会得到 $\tfrac00$。导数是 $h\to0$ 时的极限,你要先化简差商(约去 $h$),再令 $h\to0$。跳过极限是经典的错误。
From the definition, $f(x)=x^2$ gives $f'(x)=2x$. Evaluate $f'(3)$. · 根据定义，$f(x)=x^2$给出$f'(x)=2x$。计算$f'(3)$。
$f'(3)=2\cdot3=6$.
Use the definition to find $f'(x)$ for $f(x)=x^2$.
- $\dfrac{f(x+h)-f(x)}{h}=\dfrac{(x+h)^2-x^2}{h}=\dfrac{x^2+2xh+h^2-x^2}{h}$.
- $=\dfrac{2xh+h^2}{h}=2x+h$ (for $h\neq0$).
- $\displaystyle\lim_{h\to0}(2x+h)=2x$, so $f'(x)=2x$.
用定义求 $f(x)=x^2$ 的 $f'(x)$。
- $\dfrac{f(x+h)-f(x)}{h}=\dfrac{(x+h)^2-x^2}{h}=\dfrac{x^2+2xh+h^2-x^2}{h}$。
- $=\dfrac{2xh+h^2}{h}=2x+h$(当 $h\neq0$)。
- $\displaystyle\lim_{h\to0}(2x+h)=2x$,所以 $f'(x)=2x$。
The derivative $f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}$ (or the point form $f'(a)=\lim_{x\to a}\frac{f(x)-f(a)}{x-a}$) is the limit of a difference quotient — the slope of the tangent line, i.e. the instantaneous rate of change. Write it as $f'(x)$, $\frac{dy}{dx}$, or $\frac{d}{dx}[f(x)]$.
导数 $f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}$(或点形式 $f'(a)=\lim_{x\to a}\frac{f(x)-f(a)}{x-a}$)是差商的极限——即切线斜率,也就是瞬时变化率。可写作 $f'(x)$、$\frac{dy}{dx}$ 或 $\frac{d}{dx}[f(x)]$。