Suppose we have a function $v$ of $x$ with a minimum at $x=0$. We have, for $x$ close to zero, $$v'(x) = v'(0) +xv''(0) +\frac{x^2}{2}v'''(0)+\cdots$$ Then as $v'(0)=0$ $$v'(x)\approx xv''(0)$$ if $$|xv'''(0)|\ll v''(0)$$
Which is fine. I am unable to understand this statement:
Typically each extra derivative will bring with it a factor of $1/L $ where $L$ is the distance over which the function changes by a large fraction. So $$x\ll L$$
This is extracted from a physics derivation, and I cannot get how they tacked on a factor of $1/L$