File:Detachment derivative dy.png

English: The detachment and the derivative operators are both fundamental operators, naturally defined via the term dy. Note that if we immediately apply the limit process to the term ${\displaystyle \Delta y=f(x+h)-f(x)}$ then the result is a non-informative operator, since it is zeroed for continuous functions. The derivative is capable of supplying an interesting information regarding the function f by comparing between the term ${\displaystyle \Delta y}$ and the term ${\displaystyle \Delta x}$ before applying the limit process, via a fraction. The detachment relies on a similar algorithm: it compares between the term ${\displaystyle \Delta y}$ and a constant (zero) via the sgn function. Thus, the detachment is satisfied with less information than the derivative - and this fact causes the detachment to have advantages and disadvantages, depending on the context.