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?

smileycellist2

smileycellist2

Answered question

2020-12-12

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?

Answer & Explanation

Neelam Wainwright

Neelam Wainwright

Skilled2020-12-13Added 102 answers

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=2fx2=x(fx)
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=2fyx=y(fx)
When, the function is differentiated firstly with y and then with x is can be denoted as fyx:
fyx=2fxy=x(fy)
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.

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