Big Derivative

Start by constructing the Taylor series for $x^3\sin(x^2)$ centered at $x = 0$ . $x^3\sin(x^2) = x^3\sum_{n = 0}^{\infty} (-1)^n \frac{(x^2)^{2n + 1}}{(2n + 1)!} = \sum_{n = 0}^{\infty} (-1)^n \frac{x^{4n + 5}}{(2n + 1)!}$

Note that, at $x = 0$ , all terms in the power series having a power less than 25 will become 0 by virtue of differentiating them 25 times. Furthermore, all terms in the power series having a power larger than 25 will become 0 by virtue of differentiating them 25 times and plugging in 0. However, the term corresponding to $x^{25}$ will not vanish after taking 25 derivatives and plugging in 0. Rather, the 25th derivative of $x^{25}$ is $25!$ . We examine the term in the Taylor series corresponding to $n = 5$ , which is $-\frac{x^{25}}{11!}$ . We take its derivative 25 times and obtain $-\frac{25!}{11!}$ .