# 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?