Mixed Partial Derivatives If f is a function of a and y such that fxyandfyx are continuous, what is the relationship between the mixed partial derivatives?

Question

Neelam Wainwright · Accepted Answer

Given: If f is a function of xand ysuch that fxyandfyx are continuous. Explanation: Consider f is function of (x, y) such that fxyandfyx are continuous, Consider z = f (x,y). There are four ways to determine second-order partial derivatives. If is differentiated partially with respect to same dependent variable then derivatives will be fxxandfyy. It is shown below fxx=∂2f∂x2=∂∂x(∂f∂x) Differentiate partially twice with respect to y. A derivative is called mixed derivative function when a function is differentiated twice, but with different variables. They are of two types: When, the function is differentiated firstly with x and then with yis can be denoted as fxy: fxy=∂2f∂y∂x=∂∂y(∂f∂x) When, the function is differentiated firstly with y and then with x is can be denoted as fyx: fyx=∂2f∂x∂y=∂∂x(∂f∂y) In both cases, the results obtained would be the same and equivalent. Hence, both mixed partial derivatives are said to be equal, that is, fxy=fyx.

Mixed Partial Derivatives If f is a function of a and y such that f_(xy) and f_(yx) are continuous, what is the relationship between the mixed partial derivatives?

Answered question

Answer & Explanation

New Questions in Differential Calculus