Find higher order derivatives with DSolve

Questions
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1

When I have a differential equation like

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},
 x[t], t]

I can easiliy solve for x[t]. When I look for x’[t], x’'[t] or some higher order derivative I would have to rewrite the expression by moving the d#‘s one line down so x[t] becomes x’[t], but this generates a lot of code.

Is there a command for getting x’[t] and so on directly? I didn’t find a hint in the documentation… With NDSolve it is easy, but with DSolve I’m stuck.

2

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
3

Thanks, I didn’t know that. I just realized that then it’s also possible to have Sqrt’‘’'[x] etc., Problem fixed.