If $1/(N-1)>x\geq 1/N$, then $p_N:=(Nx)^{(N+1)x^{\dots}}=\infty$, $p_{N-1}=0$, $p_{N-2}=1$, and so we have a tower of height $N-3$. Thus your function has a jump in points $1/N$ and is smooth between consecutive jumps.
How to differentiate a tower? Simply fix all but one appearings of $x$, differentiate with respect to this $x$, and then sum up.