partial derivatives

Complete

Summary

Higher order partial derivatives

Clairaut’s theorem

  • applies to most functions

Equation of tangent plane

Chain rule for partial derivatives

chain_rule_partial.png

Implicit differentiation theorem

Concept

Partial derivatives

  • focus on bivariate functions for #ma1521
  • do the small change over each variable individually
  • treate the other variable as a constant

Equation of tangent plane

General form of the chain rule

  • for n variable function u, which takes in m variable functions x

chain_rule_partial_2.png

Increments

change in resulting from changes in and

Differentials

change in the tangent plane, gives a good approximation of increment over smaller changes

Application

Computing partial derivatives

Implicit differentiation with partial derivatives

Increment