differential of a real function

微分

Analysis, Measure & PDE · Undergraduate · Mathematics Resources editorial compilation (AI-assisted)

Translation evidence: Literature usage

Meaning in context

For a function differentiable at a, its differential is the linear map h mapped to f'(a)h in f(a+h)=f(a)+f'(a)h+o(|h|).

实函数在 a 处可微时，展开式中把增量 h 映为 f'(a)h 的线性映射称为微分，其余项相对于 |h| 趋于零。

Example

For $f(x)=x^2$ at $a=3$, $df_3(h)=6h$.

Usage and distinctions

The differential is the linear approximation, not the exact finite increment.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

1. Michael Taylor — Introduction to Analysis in One Variable§4.1, equation (4.1.3) · Checked 2026-10-14

Translation sources

1. 中国矿业大学数学学院 — 数学分析讲义，第五章§5 一、微分的概念, 定义1 and 定理1 · Checked 2026-10-14

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