  In this article, for some $d-$dimensional Gaussian processes
  \[X=\big\{X_t=(X^1_t,\cdots,X^d_t):t\ge0\big\},\]
  whose components are i.i.d. $1-$dimensional self-similar Gaussian process
with Hurst index $H\in(0,1)$, we consider the asymptotic behavior of
approximation of its $\boldsymbol{k}-$th derivatives of local time under
certain mild conditions, where
  $\boldsymbol{k}=(k_1,\cdots,k_d)$ and $k_\ell$'s are non-negative real
numbers. We will give a derivative version of the limit theorems for functional
of Gaussian processes and use this result to get the asymptotic behaviors.
