Ghioca.

Problem.
Find (arctan(x))(2007)(0) (i.e. the 2007th derivative of
arctan(x) at x = 0).
There is no way to keep differentiating arctan(x) and hope to see
some pattern. The derivatives get soon very ugly. Instead, we use
the already computed power series for arctan(x):
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My calculus prof writes his lecture slides exactly like he speaks. Just say the above with an Italian accent and throw your arms about Italian-like, and there you go.