Inverse Laplace transforms to maxima. How to convert a formula containing a differential?

Question

Mathematics

There was a question about using Maxima.
Maxima has functions for performing the Laplace transform (laplace) and, accordingly, the inverse Laplace transform (ilt)

(%i16) 	c:'diff(x(t),t);  
(%i17) 	atvalue(x(t),t=0,0);  
(%i18) 	atvalue(c,t=0,0); 
(%i19) 	z:laplace(c,t,p); 
(%i20) 	ilt(z,p,t);

Basically direct conversion works.
so for example the result of z:laplace(c,t,p);
will transform dx(t)/dt into p* laplace(x(t),t,p),
which looks similar to the required p*x(p)
The ilt function does not cope with the inverse Laplace transform.
issuing ilt(p*laplace(x(t),t,p),p,t)
Are there any solutions?
I've looked all over the internet, but I haven't found a solution to this problem.
Although I most likely simply do not understand anything either in mathematics or in Maxima
and I am waiting for something wrong. But in theory, the inverse Laplace transform should return something similar to diff(x(t),t);?