Question

is there exist inverse of y(t)=dx(t)/dt ?

Answer 1

depends what you mean.
there is dt/dx which gives dx/dt * dt/dx= 1
https://en.wikipedia.org/wiki/Inverse_functions_and_differentiation

and there is integration as the inverse of differentiation
http://www.kpelletij-iitjee.com/Maths/Integrals/integration-inverse-process-of-differentiation.aspx

Answer 2

the above mentioned wiki page gives you a very deep idea but one should remember the term "inverse" doesn't always mean 1/(given function)..
sin-1(x) is not equal to 1/sin(x)