  The recent discovery of a Higgs boson at the LHC, while establishing the
Higgs mechanism as the way of electroweak symmetry breaking, started an era of
precision measurements involving the Higgs boson. In an effective Lagrangian
framework, we consider the $e^+e^-\rightarrow ZHH$ process at the ILC running
at a centre of mass energy of 500 GeV to investigate the effect of the $ZZH$
and $ZZHH$ couplings on the sensitivity of $HHH$ coupling in this process. Our
results show that the sensitivity of the trilinear Higgs self couplings on this
process has somewhat strong dependence on the Higgs-gauge boson couplings.
Single and two parameter reach of the ILC with an integrated luminosity of 1000
fb$^{-1}$ are obtained on all the effective couplings indicating how these
limits are affected by the presence of anomalous $ZZH$ and $ZZHH$ couplings.
The kinematic distributions studied to understand the effect of the anomalous
couplings, again, show a strong influence of $Z$-$H$ couplings on the
dependence of these distributions on $HHH$ coupling. Similar results are
indicated in the case of the process, $e^+e^-\rightarrow \nu\bar \nu HH$,
considered at a centre of mass energy of 2 TeV, where the cross section is
large enough. The effect of $WWH$ and $WWHH$ couplings on the sensitivity of
$HHH$ coupling is clearly established through our analyses in this process.
