How do we take the first derivative with a convolution?  Let y = f(x).  Recall that y’ = lim dx->0 of (f(x+dx) – f(x))/dx.  In a discrete image, the smallest we can make dx is 1 pixel, so we can take f(x+1)-f(x), which is correlation with a filter of [-1 1].  This is asymmetric, so it’s also reasonable to say: y’=lim dx->0 of (f(x+dx) – f(x-dx))/2dx, which leads to a filter of [-.5 0 .5].  In the limit, these are the same, but before that they are different.