Basics of Differentials

Differentials capture the linear approximation of a function near a point, turning derivative information into a small-increment language. This section complements the derivative basics and focuses on definitions, properties, and quick applications.

Definition

Differential

If $f(x)$ is differentiable at $x$, its differential at that point is

$$df = f'(x)\,dx,$$

where $df$ is the change in $f$ and $dx$ is a small change in the independent variable.

Form of the Differential

Differential formula

$$df = f'(x)\,dx$$

To compute a differential, find the derivative first, then multiply by $dx$.

Geometric meaning

On the graph of $y = f(x)$, $df$ is the vertical change along the tangent line produced by a horizontal step $dx$. The pair $(dx, df)$ is the linearized version of the curve near the point.

Properties

Linearity

Linearity

$$d(af + bg) = a \cdot df + b \cdot dg$$

Product rule

Product rule

$$d(uv) = u \cdot dv + v \cdot du$$

Quotient rule

Quotient rule

$$d\!\left(\frac{u}{v}\right) = \frac{v \cdot du - u \cdot dv}{v^2}$$

Quick uses

1. Linear approximation: $f(x + \Delta x) \approx f(x) + df$ for small $\Delta x$.
2. Error estimation: bound output error by controlling $|dx|$.
3. Change of variables: rewrite expressions via $dx$ and $df$ when substituting.

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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
$dx$	数学符号	“dee x”	自变量的微小增量
$df$	数学符号	“dee f”	函数值的微小增量

中英对照

中文术语	英文术语	音标	说明
微分	differential	/ˌdɪfəˈrɛnʃəl/	将导数转化为增量形式的概念
线性性	linearity	/ˌlɪniˈærəti/	微分对加法与常数倍运算保持线性
乘积法则	product rule	/ˈprɒdʌkt ruːl/	微分的乘积拆分公式
商法则	quotient rule	/ˈkwəʊʃənt ruːl/	微分的商函数公式
线性近似	linear approximation	/ˈlɪniə əˌprɒksɪˈmeɪʃən/	使用微分估计函数值变化
误差估计	error estimation	/ˈɛrə ˌɛstɪˈmeɪʃən/	利用微分评估计算误差
变量替换	change of variables	/tʃeɪndʒ əv ˈvɛəriəb(ə)lz/	基于微分进行的变量转换

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