What are the equations of motion for a time dependent Lagrangian?

cheeseits

Let's say that L=((1/2)m*v^2-V(x))*f(t), or something similar. What are the equations of motion? For time independent it should be: (d/dt) (dL/dx_dot)=dL/dx .
Using this I get m[tex]\ddot{x}[/tex]+m f_dot/f x_dot+dV/dx=0.
Is this right? I keep thinking about the derivation of the equations and it seems like there should be a time varying term. When the action is varied, there is a term from x, x_dot, and t, right? Then integrate by parts, and factor out the dx term, argue that the integrand must be zero. Where does the time part get lost or where does it show up?

Thanks.