[tex]
\frac{d v}{d t} = \lim_{\Delta t \rightarrow 0} \frac{\Delta v}{\Delta t} = \lim_{\Delta t \rightarrow 0} \frac{v(t + \Delta t) - v(t)}{\Delta t}[/tex]
Now, look at the numerator (angle brackets mean dimension of ...):
[tex]
\left[ v(t) \right] = \left[ v(t + \Delta t)\right] = \left[ v \right] \Rightarrow \left[ \Delta v \right] \equiv \left[ v(t + \Delta t) - v(t) \right] = \left[ v \right][/tex]
where we had used the rule from Dimensional analysis that one can only add and subtract physical quantities with the same dimension, and the result is of the same dimension.
Next, look at the fraction:
[tex]
\left[ \frac{\Delta v}{\Delta t} \right] = \frac{\left[\Delta v \right]}{\left[ \Delta t \right]} = \frac{\left[ v \right]}{\left[ t \right]}[/tex]
where we had used the rule of Dimensional analysis that the dimension of a product or ratio of two physical quantities is the product or ratio of their dimensions.
Even if you take the limit as [itex]\Delta t \rightarrow 0[/itex], the dimensions of the ratio do not change. So:
[tex]
\left[ \frac{d v}{d t} \right] = \left[ \frac{\Delta v}{\Delta t} \right] = \frac{\left[ v \right]}{\left[ t \right]}[/tex]
This is a general rule: The dimensions of a derivative of y w.r.t. x is simply [y]/[x].
