taylor series expansion question

I'm having trouble getting a Taylor series expansion of an expression
involving the derivative.  I have the equation f(t) = (d/dt)u(t)+a*u(t)
and I would like to obtain the Taylor series expansion of f(t+dt) (or
f(t+2dt)) about the point t.

So far, the closest I have come is by using the following;

fo:taylor('diff(u(t),t)+a*u(t),t,0,4);

Which gives me the expansion of f(t) around t=0 and I can work with that. 
Then I am stumped with how to get the Taylor series expansion of f(2*t)
around t=0.  Something like

fo:taylor('diff(u(2*t),t)+a*u(2*t),t,0,2)

really doesn't work.

Thanks.

Todd