Inspired by Vieta's Derivatives

Calculus Level 1

The function \[f(x) = x^5 + 6x^4 - 18x^3 - 10x^2 + 45x - 24\] has only four distinct roots, each of which is real. Let the four roots be \(\alpha,\ \beta,\ \gamma,\) and \(\delta,\) in no particular order. Also let \(f'(x)\) denote the first derivative of \(f(x).\) Evaluate \[f'(\alpha)\times f'(\beta)\times f'(\gamma)\times f'(\delta).\]

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