I understand big O notation in computational complexity theory, but I don't see how it applies in the equation below.From Pattern Recognition and Machine Learning:If we weren't familiar with the rules of ordinary calculus, we could  evaluate a conventional derivative $dy/dx$ by making a small change   to the variable $x$ and then expanding in powers of $\epsilon$, so that  $y(x + \epsilon) = y(x) + \frac {dx}{dy}\epsilon + O(\epsilon^2)$and finally taking the limit $\lim\epsilon\to 0$.
What does does $O$ mean in this context?
asymptotics;machine learning;notation
It means exactly the same thing.  It means there is a constant $c$ such that the term is $\le c \cdot \epsilon^2$ (where $c$ does not depend on $\epsilon$).  Big-O notation is not unique to computer science; it is also widely used in mathematics.  See https://en.wikipedia.org/wiki/Big_O_notation, which explains some ways it is used in mathematics.