Lecture Note
University
Massachusetts Institute of TechnologyCourse
Multivariable CalculusPages
1
Academic year
2022
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.
Exploring Partial Derivatives and Their Definitions
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