  We demonstrate that a method using the derivative of the selective reflection
signal from a nanocell is a convenient and robust tool for atomic laser
spectroscopy, achieving a nearly Doppler-free spectral resolution. The recorded
linewidth of the signal from a potassium-filled cell, whose thickness $\ell$
lies in the range $350-500$ nm, is 18 times smaller than the Doppler linewidth
($\sim 900$ MHz full width at half maximum) of potassium atoms. We also show
experimentally a sign oscillation of the reflected signal's derivative with a
periodicity of $\lambda/2$ when $\ell$ varies from 190 to 1200~nm confirming
the theoretical prediction. We report the first measurement of the van der
Waals atom-surface interaction coefficient $C_3 = 1.9\pm 0.3$
kHz$\cdot\mu$m$^3$ of potassium $4S_{1/2} \rightarrow 4P_{3/2}$ transitions
with the nanocell's sapphire windows, demonstrating the usefulness and
convenience of the derivative of selective reflection technique for cell
thicknesses in the range $60 -120~$nm.
