http://calc3.askbot.com/questions/Open source question and answer forum written in Python and DjangoenCopyright Askbot, 2010-2011.Tue, 08 Jul 2014 12:01:49 -0500http://calc3.askbot.com/question/62/how-to-find-mixed-partial-derivatives/On the in class worksheet, question number five asks to show by example that the mixed partial derivatives of f(x,y)=$x^3y^4$ are equal. I understand how to take the partial derivatives, but what is meant by *mixed* partial derivatives?Tue, 08 Jul 2014 10:02:22 -0500http://calc3.askbot.com/question/62/how-to-find-mixed-partial-derivatives/http://calc3.askbot.com/question/62/how-to-find-mixed-partial-derivatives/?answer=63#post-id-63Searching through the textbook, I found this piece of information on p. 371 (14.6): > “mixed partials” [are] computed by taking partial derivatives with respect to both variables in the two possible orders They are notated $f_{xy}$ and $f_{yx}$. To find $f_{xy}$, we take the $x$ partial derivative then the $y$ partial derivative of $f$: $$ f_x = 3x^2y^4 $$ $$ f_{xy} = (3x^2)(4y^3) = 12x^2y^3$$ Note that we leave the $x$'s alone in the second step and only focus on the $y$'s. We will use a very similar process to find $f_x{yx}$: $$ f_y = 4y^3x^3 $$ $$ f_{yx} = (4y^3)(3x^2) = 12x^2y^3 $$ $$ f_{xy} = f_{yx} = 12x^2y^3 $$ Theorem 14.6.2 (Clairaut's Theorem) then says: > If the mixed partial derivatives are continuous, they are equal. Which is a more generalized version of what we just showed.Tue, 08 Jul 2014 12:01:49 -0500http://calc3.askbot.com/question/62/how-to-find-mixed-partial-derivatives/?answer=63#post-id-63