Originally Posted by

**HallsofIvy** All I can say is that the "answer", $\displaystyle 2x- xsiny- 2y/(x- y^2)^2$ is **wrong**.

You can do this either of two ways. Since you were asked to find $\displaystyle f_y$ and had already found that $\displaystyle f_y= x^2+ x cos(y)+\frac{2y}{x-y^2}$ just differentiate that with respect to x: $\displaystyle f_{yx}= 2x+ cos(y)- \frac{2y}{(x- y^2)^2}$.

Or, first find $\displaystyle f_x= 2xy+ siny- \frac{1}{x-y^2}$ and differentiate that with respect to y: $\displaystyle f_{xy}= 2x+ cos y- \frac{2y}{(x- y^2)^2}$.

(As long as the second derivatives are continuous, the "mixed derivatives", $\displaystyle f_{xy}$ and $\displaystyle f_{yx}$, are equal.)