  Very recently, a new concept called multiplicative differential (and the
corresponding $c$-differential uniformity) was introduced by Ellingsen
\textit{et al} in [C-differentials, multiplicative uniformity and (almost)
perfect c-nonlinearity, IEEE Trans. Inform. Theory, 2020] which is motivated
from practical differential cryptanalysis. Unlike classical perfect nonlinear
functions, there are perfect $c$-nonlinear functions even for characteristic
two. The objective of this paper is to study power function $F(x)=x^d$ over
finite fields with low $c$-differential uniformity. Some power functions are
shown to be perfect $c$-nonlinear or almost perfect $c$-nonlinear. Notably, we
completely determine the $c$-differential uniformity of almost perfect
nonlinear functions with the well-known Gold exponent. We also give an
affirmative solution to a recent conjecture proposed by Bartoli and Timpanella
in 2019 related to an exceptional quasi-planar power function.
