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Basic Concepts of the Derivative

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

Average vs. instantaneous rate

Average rate on [x0,x0+Δx][x_0, x_0+\Delta x]:

ΔyΔx=f(x0+Δx)−f(x0)Δx.\frac{\Delta y}{\Delta x} = \frac{f(x_0+\Delta x) - f(x_0)}{\Delta x}.

Instantaneous rate (the derivative):

f′(x0)=lim⁡Δx→0f(x0+Δx)−f(x0)Δx.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)=x2f(x)=x^2:

Formal definition

Derivative at a point

If

lim⁡Δx→0f(x0+Δx)−f(x0)Δx\lim_{\Delta x \to 0} \frac{f(x_0+\Delta x) - f(x_0)}{\Delta x}

exists, ff is differentiable at x0x_0, and the limit is f′(x0)f'(x_0) (also written dydx∣x0\dfrac{dy}{dx}\big|_{x_0}).

Why dydx\dfrac{dy}{dx}?

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

Equivalent forms

f′(x0)=lim⁡h→0f(x0+h)−f(x0)horf′(x0)=lim⁡x→x0f(x)−f(x0)x−x0.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−′(x0)=lim⁡h→0−f(x0+h)−f(x0)hf'_-(x_0) = \lim_{h\to 0^-} \dfrac{f(x_0+h)-f(x_0)}{h}
  • Right: f+′(x0)=lim⁡h→0+f(x0+h)−f(x0)hf'_+(x_0) = \lim_{h\to 0^+} \dfrac{f(x_0+h)-f(x_0)}{h}

Differentiable at x0x_0 ⇔ both exist and are equal.

Differentiability vs. continuity

定理

If ff is differentiable at x0x_0, then ff is continuous at x0x_0.

几何解释
推论
证明
  1. Differentiability gives the limit of the difference quotient.
  2. Multiply by hh and send h→0h\to 0 to see f(x0+h)→f(x0)f(x_0+h)\to f(x_0).
  3. Continuity does not imply differentiability (e.g., ∣x∣|x| at 0).
符号说明

Examples and practice

Example: f(x)=x2f(x)=x^2

f′(x0)=lim⁡Δx→0(x0+Δx)2−x02Δx=2x0.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)=1xf(x) = \dfrac{1}{x} at x0x_0 using the definition.

Reference Answer(2 个标签)
derivativesdifferentials
f′(x0)=lim⁡Δx→01x0+Δx−1x0Δx=−1x02.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∣f(x)=|x| differentiable at 00?

Reference Answer(2 个标签)
derivativesdifferentials

f−′(0)=−1, f+′(0)=1f'_-(0) = -1,\ f'_+(0)=1 ⇒ not differentiable at 00.

Summary

本文出现的符号

符号类型读音/说明在本文中的含义
Δx\Delta x数学符号Delta x自变量的有限增量
dxdx数学符号“dee x”自变量的无穷小增量
dydy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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Differential Calculus of One Variable