Determination gives success!

Let $f(x)=1+2x+3x^2+4x^3+...+10x^9$ with $x_1,x_2,x_3...x_9$ as the roots of $f(x)=0$

$e_1=\sum x_1$

$e_2=\sum x_1x_2$

$e_3=\sum x_1x_2x_3$ and so on till

$e_9=\sum x_1x_2x_3...x_9$

And $\Delta = \left| \begin{array}{ccc} e_1 & e_2 & e_3 & \\ e_4 & e_5 & e_6 & \\ e_7 & e_8 & e_9 & \end{array} \right|$

Find $\Delta^{-1}$

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This is an original problem and belongs to the set My Creations .