Indeed this is a simple procedure.  You can start from the flexural vibration equation of a beam, then applying the boundary conditions (one fixed end and one free end).  You can find the complete procedure in books by Timoshenko like the Vibration one.

 Also many books on elementary vibration and structural analysis should have the procedure given.

R. Chennamsetti, Scientist, R&D Engineers, India

The procedure is as following.

(a) Derive governing equation of a beam for lateral vibration. You need to use Newton's second law. This will be fourth order in space and second order in time.

(b) Assume there is no external loading on the beam. Use separation of variables technique. We get two separate equations, one is in space and the other in temporal domain. Both equal to some constant (this is sqaure of circular frequency).

(c) In spatial domain, the equation is fourth order. Four constants appear in the solution. Use four boundary conditions of beam, two at each end.  For a cantilever (fixed-free) beam, in first mode (1.875)^2 = 3.52 (approx) appears.  

(d) In time domain, the equation is second order. Two initial conditions are required.

(e) f = {(1.875)^2/2pi}sqrt(EI/(m*L^3)) = {(1.875)^2/2pi}sqrt(EI/(rho*A*L^4))

You may check your term in the denominator, which is in square root.

For more details refer the following.

S. S. Rao, 'Mechanical Vibrations' 4th edition, Pearson Edition, Page - 609 - 613.