Difference quotient ($\frac{f(x+h)-f(x)}{h}$)

A difference quotient divides a finite change in a function by the corresponding change in its argument. Its limit, when it exists, is the derivative.

With ${\delta }_{h}f(x)=[f(x+h{e}_{\ell })-f(x)]/h$, a change of variables givesThe compact support of $f$ and its translates must remain inside the integration domain, or one must integrate over the whole space with convergent integrals. In particular ${\delta }_{-h}g(x)=[g(x)-g(x-h{e}_{\ell })]/h$: the denominator's sign matters. This is the difference quotient analogue of integration by parts.