Vieta's Derivative II -- My 500-follower problem

Calculus Level 4

The function $x^5 + 6x^4 - 18x^3 -10x^2 + 45x -24$ has only four distinct real roots: $\alpha$ , $\beta$ , $\gamma$ and $\delta$ (in no particular order). If $f'(x)$ is the first derivative of $f(x)$ . Evaluate

$f'(\alpha) + f'(\beta) + f'(\gamma) + f'(\delta)$

Inspiration: This is exactly the problem titled Inspired by Vieta's Derivatives with a twist.