Basic Concepts of the Derivative

The derivative measures an instantaneous rate of change—how fast $f(x)$ moves when $x$ moves by a tiny amount.

Average vs. instantaneous rate

Average rate on $[x_0, x_0+\Delta x]$:

$$\frac{\Delta y}{\Delta x} = \frac{f(x_0+\Delta x) - f(x_0)}{\Delta x}.$$

Instantaneous rate (the derivative):

$$f'(x_0) = \lim_{\Delta x \to 0} \frac{f(x_0+\Delta x) - f(x_0)}{\Delta x}.$$

Geometric picture

Average rate = secant slope; instantaneous rate = tangent slope. Explore this with $f(x)=x^2$:

选择点的 x 坐标： x = 1.0

函数值

f(1.0) = 1.00

瞬时变化率（导数）

f'(1.0) = 2.00

📊 观察要点：

蓝色曲线是函数 f(x) = x² 的图像

橙色虚线是在选定点的切线

红色点是切点，标签显示了该点的坐标和切线斜率（即瞬时变化率）

拖动滑块可以看到，随着 x 的变化，切线的斜率也在变化

Formal definition

Derivative at a point

If

$$\lim_{\Delta x \to 0} \frac{f(x_0+\Delta x) - f(x_0)}{\Delta x}$$

exists, $f$ is differentiable at $x_0$, and the limit is $f'(x_0)$ (also written $\dfrac{dy}{dx}\big|_{x_0}$).

Why $\dfrac{dy}{dx}$?

• $\dfrac{\Delta y}{\Delta x}$: slope of a secant (finite increments).
• $\dfrac{dy}{dx}$: slope of the tangent (infinitesimal increments).

调整 Δx (dx) 的大小： Δx = 1.00

Δy (Delta y)：函数值的实际变化量 (曲线上的高度差)

dy (微分 y)：切线上的变化量 (线性近似)

💡 观察：当 Δx 变得很小很小时，Δy 和 dy 几乎重合。这就是为什么我们可以用 dy/dx 来表示导数。

Equivalent forms

$$f'(x_0) = \lim_{h\to 0} \frac{f(x_0+h)-f(x_0)}{h} \quad\text{or}\quad f'(x_0) = \lim_{x\to x_0} \frac{f(x)-f(x_0)}{x-x_0}.$$

One-sided derivatives

• Left: $f'_-(x_0) = \lim_{h\to 0^-} \dfrac{f(x_0+h)-f(x_0)}{h}$
• Right: $f'_+(x_0) = \lim_{h\to 0^+} \dfrac{f(x_0+h)-f(x_0)}{h}$

Differentiable at $x_0$ ⇔ both exist and are equal.

Differentiability vs. continuity

定理

If $f$ is differentiable at $x_0$, then $f$ is continuous at $x_0$.

证明

1. Differentiability gives the limit of the difference quotient.
2. Multiply by $h$ and send $h\to 0$ to see $f(x_0+h)\to f(x_0)$.
3. Continuity does not imply differentiability (e.g., $|x|$ at 0).

Examples and practice

Example: $f(x)=x^2$

$$f'(x_0) = \lim_{\Delta x\to 0} \frac{(x_0+\Delta x)^2 - x_0^2}{\Delta x} = 2x_0.$$

练习 1

Differentiate $f(x) = \dfrac{1}{x}$ at $x_0$ using the definition.

参考答案

$$f'(x_0) = \lim_{\Delta x\to 0} \frac{\frac{1}{x_0+\Delta x}-\frac{1}{x_0}}{\Delta x} = -\frac{1}{x_0^2}.$$

练习 2

Is $f(x)=|x|$ differentiable at $0$?

参考答案

$f'_-(0) = -1,\ f'_+(0)=1$ ⇒ not differentiable at $0$.

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总结

本文出现的符号

符号	类型	读音/说明	在本文中的含义
$\Delta x$	数学符号	Delta x	自变量的有限增量
$dx$	数学符号	“dee x”	自变量的无穷小增量
$dy$	数学符号	“dee y”	因变量的无穷小增量

中英对照

中文术语	英文术语	音标	说明
导数	derivative	/dɪˈrɪvətɪv/	函数在点的瞬时变化率
平均变化率	average rate of change	/ˈævərɪdʒ reɪt əv tʃeɪndʒ/	有限区间的变化率
瞬时变化率	instantaneous rate of change	/ɪnstənˈteɪniəs reɪt əv tʃeɪndʒ/	极限意义下的变化率
微分	differential	/ˌdɪfəˈrɛnʃəl/	无穷小增量
可导	differentiable	/ˌdɪfəˈrɛnʃəbl/	导数存在
连续	continuous	/kənˈtɪnjʊəs/	函数无跳跃

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