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.

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