  Effects of the delta-isobar $(\Delta)$ mixing on the spin-isospin response
functions in finite nuclei are studied in the quasi-elastic region. A method to
calculate the response function for a finite system composed of nucleon $(N)$
and $\Delta$ is formulated in a ring approximation. It is designed to treat the
$\Delta$-related Landau-Migdal parameters, $g'_{N\Delta}$ and
$g'_{\Delta\Delta}$, and the nucleon parameter $g'_{NN}$, independently, so
that the universality ansatz, $g'_{NN}=g'_{N\Delta}=g'_{\Delta\Delta}$, is
removed. We calculated the isovector spin-longitudinal and -transverse response
functions, $R_{L}$ and $R_{T}$, with and without the $\Delta$-mixing. Inclusion
of $\Delta$ enhances $R_{L}$ but reduces $R_{T}$ for ordinary interactions.
Dependence of $R_{L,T}$ on $g'_{NN}$ and $g'_{N\Delta}$ is investigated.
Decomposition into the process-decomposed response functions, $R^{[NN]}$,
$R^{[N\Delta]}$ and $R^{[\Delta\Delta]}$, is very elucidative to see the
$\Delta$ effects, which are found to be mainly governed by $R^{[N\Delta]}$ and
sensitive to $g'_{N\Delta}$. The isovector spin-transverse response function
$R^{(e,e')}_{T}$ obtained by $(e,e')$ is calculated by various effective
interactions and compared to each other as well as experimental data.
