Let $g(x) = \exp(f(x))$. Assuming numerical (or symbolic) values of $f(x), f'(x), f''(x), \ldots, f^{(n)}(x)$ are known, is there a way to compute $g'(x), g''(x), \ldots g^{(n)}(x)$ (or even the single value $g^{(n)}(x)$) for large $n$ that is faster than explicitly generating and evaluating the expanded symbolic derivative, a polynomial which has $p(n)$ (partition function) terms?

Yes, you can. For example, the normal evaluation for $n=6$ requires 33 multiplications and 10 additions. But by using an optimized straightline program which takes care of common subexpression elimination, you can reduce that to 17 multiplications, 10 additions and 4 assignments. For $n=8$, the reduction in multiplications goes from 84 down to 37, and at $n=12$, it goes from 397 down to 114. Note that multiplication by constants (like $10f'(x)$) counts as a multiplication too. You can experiment with these things by using the Unfortunately, at this point, I do not immediately see a pattern in the results, so I am not sure how to automate this and produce optimized versions directly. But the answer to your original question is definitely 'yes'. 


I feel that I should post an answer, for the benefit of any potential readers from the future... For simultaneous computation of the first $n$ derivatives, the problem is really computing the exponential of a power series, and it is a classical result that this can be done in $O(M(n))$ arithmetic operations (which most likely is optimal). See: Brent and Kung, Fast algorithms for manipulating formal power series, Journal of the ACM, 1978. There are more recent papers reducing the implied constant factors. I'm still not aware of any result to the effect that the $n$th derivative alone can be computed faster than $O(M(n))$. 


Try automatic differentiation, which is exact but not symbolic. See also The Arithmetic of Differentiation by L. B. Rall. 

