Let $Φ_n^{(k)}(x)$ be the $k$-th derivative of $n$-th cyclotomic polynomial. Extending a work of D.~H.~Lehmer, we show some curious congruences: $2Φ^{(3)}_n(1)$ is divisible by $φ(n)-2$ and $Φ^{(2k+1)}_n(1)$ is divisible by $φ(n)-2k$ for $k\ge 2$. The congruence stems from a general property of self-reciprocal polynomials.