Show that for the function $$f(x,y) = 9x^3y+2y^3+10x^2y^2+9$$ satisfies the equality $$f_{yx} = f_{xy}$$ by computing the partial derivatives.
I know that $f_y= 9x^3+6y^2$ because we exclude all terms with $x$ and constants. So $f_{yx}=27x^2$. $f_x = 10y^2$ because all the terms with $y$ go away. But it does not satisfy the equality. Why not?
Just because we take the partial with respect to $x$, does not mean all the terms with $y$ go away. We just treat them as constants. So $$f_y = 9x^3+6y^2+20x^2y$$ because we treat $x$ as a constant. $$f_{yx} = 27x^2+40xy$$Now doing $f_{xy}$ $$f_x = 27x^2y+20xy^2$$ and $$f_{xy} = 27x^2+40xy$$. Thus $$f_{xy} = f_{yx}$$