The Lehmer-Comtet sequence
Haskell, 77 75 bytes, no differentiation builtins

x@(a:b)&amp;y@(c:d)=a*c:zipWith(+)(b&amp;y)(x&amp;d)
s=1:s&amp;(1:scanl(*)1[-1,-2..])
(s!!)

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How it works
We represent a function as its infinite list of Taylor series coefficients about \$x = 1\$:
$$
f(x) = \sum_{n=0}^\infty \frac{f^{(n)}(1)}{n!} (x - 1)^n
$$
is represented by \$[f(1), f'(1), f''(1), ...]\$.
The &amp; operator multiplies two such functions using the product rule.  This lets us recursively define the function \$s(x) = x^x\$ in terms of itself using the differential equation
$$
s(1) = 1, \\
s'(x) = s(x) \cdot (1 + \ln x),
$$
where
$$
\ln x = \sum_{n=1}^\infty \frac{(-1)^{n-1}(n - 1)!}{n!}(x - 1)^n.
$$