Problem of 2017 (III)

$\large{\begin{aligned}f(x)&=1+x+x^2+x^3+x^4+.............+x^{2016}+x^{2017}\\ f^1(x)&=0+1+2x+3x^2+4x^3+................+2016x^{2015}+2017x^{2016}\\ f^2(x)&=0+0+2\cdot1+3\cdot2x+4\cdot3x^2+................2016\cdot2015x^{2014}+2017\cdot2016x^{2015}\\ f^3(x)&=0+0+0+3\cdot2\cdot1+4\cdot3x+....................2016\cdot2015\cdot2014x^{2013}+2017\cdot2016\cdot2015x^{2014}\\ ----&---------------------------------------\\ ----&---------------------------------------\\ f^{2017}(x)&=0+0+0+0+..................+0+2017\cdot2016\cdot2015\cdot2014...............3\cdot2\cdot1=\boxed{2017!}\end{aligned}}$