What is the Mixed Derivative Theorem for mixed second-order partial derivatives? How can it help in calculating partial derivatives of second and higher orders? Give examples.

smileycellist2 2021-03-07 Answered
What is the Mixed Derivative Theorem for mixed second-order partial derivatives? How can it help in calculating partial derivatives of second and higher orders? Give examples.
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casincal
Answered 2021-03-08 Author has 82 answers
Step 1
Mixed Derivative theorem:" If the function f(x,y) and its partial derivatives fx,fy,fxyandfyx are all defined in any open interval (a,b) and all are continues in the interval, then fxy(a,b)=fyx(a,b)".
That is, mixed derivative theorem says that the mixed partial derivatives are equal.
Thus, there is no need of calculating all the mixed partial derivatives. Only one case is enough.
Step 2
For example consider the function f(x,y)=x3y3+x2y+y2x.
Find the first order partial derivatives as follows.
x(x3y3+x2y+y2x)=x(x3y3)+x(x2y)+x(y2x)
=3y3x2+2yx+y2
y(x3y3+x2y+y2x)=y(x3y3)+y(x2y)+y(y2x)
=3x3y2+x2+2xy
Step 3
Now, find the mixed partial derivatives as,
2yx(x3y3+x2y+y2x)=x(3x3y2+x2+2xy)
=9y2x2+2x+2y
2xy(x3y3+x2y+y2x)=y(3y3x2+2yx+y2)
=9x2y2+2x+2y
That is, 2fyx=2fxy.
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