  In recent years, many papers discuss the theory and applications of new
fractional-order derivatives that are constructed by replacing the singular
kernel of the Caputo or Riemann-Liouville derivative by a non-singular (i.e.,
bounded) kernel. It will be shown here, through rigorous mathematical
reasoning, that these non-singular kernel derivatives suffer from several
drawbacks which should forbid their use. They fail to satisfy the fundamental
theorem of fractional calculus since they do not admit the existence of a
corresponding convolution integral of which the derivative is the left-inverse;
and the value of the derivative at the initial time $t=0$ is always zero, which
imposes an unnatural restriction on the differential equations and models where
these derivatives can be used. For the particular cases of the so-called
Caputo-Fabrizio and Atangana-Baleanu derivatives, it is shown that when this
restriction holds the derivative can be simply expressed in terms of integer
derivatives and standard Caputo fractional derivatives, thus demonstrating that
these derivatives contain nothing new.
