Q:

In the equation: $a = dv/dt$ , is $dt$ the time taken to achieve that instantaneous acceleration? 

If you solve for $dt$ from $a = \frac{dv}{dt}$ , is it the time taken to to achieved that instantaneous acceleration? 
$a$ : acceleration
$v$ : velocity
$t$ : time 

A:

No, it is not. 
Suppose, a body is moving at a uniform velocity $v$, now there is no restriction on how much time it wants to remain with that same velocity.  And after sometime it can accelerate if there is a net force on it.  Now, acceleration means a rate of change in velocity and obviously it will take some time to increase (or decrease) it's velocity.  It is therefore given as $$a=\frac{\Delta v}{\Delta t}$$  This acceleration may not be uniform and in principle can change.  Therefore, you would want to know what exactly was acceleration at a particular time.  This happens when you take the limit of $\Delta t$ tending to zero, so that time duration over which change of velocity happens is as short as possible to get the most accurate result which gives the instantaneous acceleration.  This differential small time is denoted by $dt$.
I hope this makes things clearer.