Rapidly decreasing functions and convergence

I am trying to show that, for f a rapidly decreasing function, the mapping from f to f' is continuous. Since the space of RDF's is an F-space, Rudin says that I can use the closed graph theorem for F-spaces, and hence I need to show that the graph of the mapping is closed.

So I wish to show that $\displaystyle \{(f, f') : f \in RDF's\}$ is closed. I think that I can do this if it follows that if $\displaystyle f_k \to f$, then $\displaystyle f_k' \to f'$?

Can someone tell me if this is true, and if so, how would I prove it?