dx=a(x)dt+b(x)dw is given for x(t).

Want to know the dynamics of f(x). the change of f(x) over a small time interval dt can be calculated as

df(x)=f(x+dx)-f(x)
=f'(x)dx+0.5f''(x)(dx)(dx) +o(dt)
=f'(x)[a(x)dt+b(x)dw]+0.5f''(x)[(adt+bdw)(adt+bdw)]+o(dt)
=f'(x)[a(x)dt+b(x)dw]++0.5f''(x)[bb(dw)(dw)]+o(dt)
now use (dw)(dw)=dt
=f'(x)[a(x)dt+b(x)dw]++0.5f''(x)[bb]dt

The last term is the Ito term.