partial derivative

偏导数

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

Translation evidence: Literature usage

Meaning in context

For f on an open subset of R^n, its jth partial derivative at a is the finite limit [f(a+h e_j)-f(a)]/h as nonzero h tends to zero, when that limit exists.

定义在欧氏空间开集上的函数，固定其他坐标，仅沿第 j 个坐标方向的增量商若有有限极限，该极限为相应偏导数。

Example

For f(x,y)=x2yf(x,y)=x^2y, ∂xf(2,3)=2⋅2⋅3=12\partial_xf(2,3)=2\cdot2\cdot3=12.

Usage and distinctions

Existence of every partial derivative at one point does not by itself imply total differentiability there.

Recommended reading in this subject · Preparation & skip guidance

Definition sources

  1. Michael Taylor — Introduction to Analysis in Several Variables (Advanced Calculus)§1.1, equation (1.1.38), printed p.10 · Checked 2026-10-14

Translation sources

  1. 北京大学数学科学学院 — 数学分析（实验班）课程介绍第二部分，多元微积分：二、多元函数微分学，偏导数 · Checked 2026-10-14
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