Perhaps
In[1]:= Clear[x]
In[2]:= x[t_] = x[t] /. Apart[DSolve[{x'''''''[t]==d7, x''''''[0]==d6, x'''''[0]==d5, x''''[0]==d4, x'''[0]==d3, x''[0]==d2, x'[0]==d1, x[0]==d0}, [t], t]][[1]]
Out[2]= d0 + d1 t + (d2 t^2)/2 + (d3 t^3)/6 + (d4 t^4)/24 + (d5 t^5)/120 + (d6 t^6)/720 + (d7 t^7)/5040
In[3]:= x'[t]
Out[3]= d1 + d2 t + (d3 t^2)/2 + (d4 t^3)/6 + (d5 t^4)/24 + (d6 t^5)/120 + (d7 t^6)/720
In[4]:= x'''''[t]
Out[4]= d5 + d6 t + (d7 t^2)/2