  In this paper, we first present a simpler proof of a result on the strict
Fr\'echet differentiability of the metric projection operator onto closed balls
centered at the origin in Hilbert spaces, which given by Li in \cite{Li24}.
Then, based on this result, we prove the strict Fr\'echet differentiability of
the metric projection operator onto closed balls with center at arbitrarily
given point in Hilbert spaces. Finally, we study the strict Fr\'echet
differentiability of the metric projection operator onto the second-order cones
in Euclidean spaces.
