I like to use little helper function called rhs, which returns the right hand side of an equation for these things.
ClearAll[x, y, v, t, eq];
rhs[eq_] := eq /. (lhs_) == (rhs_) :> rhs;
eq1 =x'[t] == x[t] y[t] + 2 x[t];
eq2 = y'[t] == y[t]^2 + 3;
eq3 = v[t] == x[t]^2 + y[t]^2;
eq4 = D[eq3, t];
which gives
v'[t] == 2 x[t] x'[t] + 2 y[t] y'[t]
and now
eq4 /. {x'[t] -> rhs[eq1], y'[t] -> rhs[eq2]}
which returns
v'[t] == 2 x[t] (2 x[t] + x[t] y[t]) + 2 y[t] (3 + y[t]^2)