Will you differentiate till you reach 60!

$p(x)=\displaystyle n\sum_{n=1}^{100} x^{n} +1 \\ p^{1}(x)=\displaystyle n\sum_{n=1}^{100} nx^{n-1} \\ p^{2}(x)=\displaystyle n\sum_{n=1}^{100} n(n-1)x^{n-2} \\ \text{From the pattern , the 60th derivative will be } \\ p^{60}(x)=\displaystyle n\sum_{n=1}^{100} n(n-1)\cdots (n-59)x^{n-60} \\ \text{Sum of All roots } = -\dfrac{\text{coefficient of } x^{39}}{\text{coefficient of } x^{40}} \\ S= - \dfrac{99\times\cancel{99}\times\cancel{98}\cdots \times 40}{100\times 100\times \cancel{99}\times \cancel{98} \cdots \times \cancel{41} } = \boxed{-\dfrac{99}{250}}$