Exploring Partial Derivatives and Their Definitions

University
Massachusetts Institute of Technology
Course
Multivariable Calculus
Pages

1

Academic year

2022
Author

Sporkz
Views

41

p {margin: 0; padding: 0;} .ft00{font-size:20px;font-family:Arial;color:#000000;} .ft01{font-size:18px;font-family:ArialMT;color:#000000;} .ft02{font-size:18px;line-height:23px;font-family:ArialMT;color:#000000;} Level curves and partial derivatives. Partial derivatives: definitions Partial derivatives are a notation used to express the derivative of a function withrespect to one of its variables, holding other variables constant. A function of severalvariables may have partial derivatives with respect to each variable, but it does nothave a derivative in the usual sense. This symbol is a capital d with a curly line on the bottom. It is not a straight line and itis not a letter d. It is referred to as a "partial differential" or a "del" for short. The partial derivative of a function f with respect to one of its variables, x , at thepoint (x0,y0), is defined by the limit of this ratio as the change in x becomesinfinitesimally small.So here we are actually not changing y at all. We are just changing x and looking atthe rate of change with respect to x. And we have the same with respect to y, partial fpartial y is the limit, so we should say at the point (x0, y0) is the limit as delta y turnsto 0. A partial derivative is a derivative of a function with respect to one of its variables,but not the others. A function is differentiable if these partial derivatives exist.So most functions are differentiable, and we will learn how to compute their partialderivatives without having to use the usual methods.