# Derivative of a vector expression?

I'm trying to find the time derivative of the following function, where E is a constant, spatially uniform vector field, and B is as well, but B varies with time.

$$\frac{d}{dt}\left(\frac{\vec{E}\times\vec{B}}{B^2}\right)$$

Remembering that B is time dependent and E is not, I've calculated the derivative of this as (where a dot above the letter indicates a time derivative):

$$\frac{1}{B^4}\left[B^2(\vec{E}\times\dot{\vec{B}})-2B\dot{B}(\vec{E}\times\vec{B})\right]$$

Can anyone tell me if this is correct?