Partials of a homogeneous cubic quotient

Consider the function $f: \mathbb{R}^2 \to \mathbb{R}$ defined by $f(x,y) = \begin{cases} \frac{x^3 - y^3}{x^2 + y^2} & \text{if } (x,y) \ne (0,0) \ 0 & \text{otherwise.} \end{cases}$ If $f_x$ denotes the partial derivative of $f$ with respect to $x$ and $f_y$ denotes the partial derivative of $f$ with respect to $y$, then which one of the following statements is NOT true?