Derivatives of null-space

Question

I have a feeling that this is a rather silly question. If I have an $r$-ranked matrix ${\bf{A}} \in {\Bbb{R}^{m \times n}}$ which describes an overdetermined system (there are more equations than unknowns, $m>n$). There exists a basis of null-space of ${\bf{A}}$, denoted as ${\bf{C}} \in {\Bbb{R}^{n \times r}}$, such that ${\bf{A}}\left( {{\bf{C\beta }}} \right) = 0$ for any ${\bf{\beta }} \in {\Bbb{R}^r}$. Matrix ${\bf{A}}$ itself is a nonlinear function of vector ${\bf{x}} \in {\Bbb{R}^{m}}$. Is there a plausible way to calculate the first derivative of ${\bf{C}}$ with respect to ${\bf{x}}$, or ${\nabla _{\bf{x}}}{\bf{C}}$?