Find the derivative of the root of cubic equation

Consider the following cubic equation:

begin{align}
alpha ^3-2 alpha ^2- alpha(6+4x^2) +8 x^3-4x&=0
end{align}

Where $xin mathbb{R}$.

The solutions $alpha_1,alpha_2,alpha_3$ will depend on $x$. Solving this equation on Mathematica or Wolfram Alpha gives me complicated third roots.

I am not interested in the actual value of the roots, I am only looking for the value of $frac{mathrm{d}alpha}{mathrm{d}x}$. Does a method exist to obtain a “nice” expression for $frac{mathrm{d}alpha}{mathrm{d}x}$ without having to actually solve the cubic equation?