I can't say I share Littlewood's sense of humor, but here is a formula that made me grin the first time I saw it ($A$ and $B$ are non-commuting square matrices):
$$\frac{d}{dt}e^{A+B t}=\int_0^1 e^{s(A+B t)}\ B\ e^{(1-s)(A+B t)} \ ds.$$