A calculus problem by Dylan Scupin-Dursema

In general, if a polynomial of degree $n$ is differentiated $n$ times, then by the power rule, it gives a polynomial of degree $(n - n) = 0,$ which is a constant. So if it is differentiated one more time, it will be $0,$ as the derivative of a constant is always $0.$ That is, $\frac{d^{n+1}}{dx^{n+1}}[P(x)] = 0$ if $P(x)$ is a polynomial of degree $n.$ In this case, $n=101$ , and from the formula above, $\frac{d^{102}}{dx^{102}}[x^{101}] = \boxed{0}$