Find a value $c$ such that $\left\|\begin{pmatrix} x^2 - y^2\\2xy \end{pmatrix}\right\| \leq |c|\left\|\begin{pmatrix} x\\y \end{pmatrix}\right\|$ This is part of a larger problem where I am trying to find the derivative of a vector valued function. I feel like I'm missing something simple. NOTE: $c$ can be a function of $x$ and $y$.

We compute
$$(x^2 - y^2)^2 + (2xy)^2 = x^{4} - 2x^{2}y^{2}+ y^{4} + 4x^{2}y^{2} = x^{4} + 2x^2 y^2 + y^4 = (x^2 + y^2)^2$$
Hence
$$\sqrt{(x^2 - y^2)^2 + (2xy)^2} = \sqrt{(x^2 + y^2)^2} = x^2 + y^2$$
We are trying to find $c$ for which
$$ (x^2 + y^2) \leq c \sqrt{x^2 + y^2}$$
If we let $z = x^2 + y^2$ we obtain
$$ z \leq c \sqrt{z}$$
Does this seem possible?
