The change in allele frequency over one generation is:
p = p[t+1] - p[t] = (1+s) p / (1 + s p) - p = (p + s p - p - s p2) / (1 + s p)
= s p (1-p) / (1 + s p)
If we call p f(s), note that:
f'(s)= p (1-p) / (1 + s p) - s p2 (1-p) / (1 + s p)2,
so f'(0) = p (1-p)
Putting these terms in the Taylor Series gives: f(s) = f(0) + s f'(0) = s p (1-p). Therefore, if s is small, p is approximately s p (1-p).
The maximum rate of change will occur at p=1/2. We can tell that a maximum or a minimum must occur at p=1/2 since at this point, d(p)/dp=0. We can tell it is a maximum since the double derivative of p with respect to p is negative (which implies a concave function).