How fast do Schwartz seminorms typically grow as the number of derivatives increases?

Question

If $f\in\mathcal S(\mathbb R)$, then $$\sup_{m\leq k,x\in\mathbb R^d}(1+|x|)^k|\partial^mf(x)|=c_k<\infty.$$ Trying a few examples numerically, $c_k$ seems to grow exponentially as $k\to\infty$. I'd appreciate any direction about how to prove asymptotics for particular functions or general results for the Schwartz class.