| ƒ″(x) = lim.h=0 | ƒ(x + h) − 2ƒ(x) + ƒ(x − h) | . |
| h2 |
(i.)
The limit expressed by the right-hand member of this equation may exist in cases in which ƒ′(x) does not exist or is not differentiable. The result that, when the limit here expressed can be shown to vanish at all points of an interval, then ƒ(x) must be a linear function of x in the interval, is important.
The relation (i.) is a particular case of the more general relation
ƒ(n)(x) = lim.h=0 h−n [ ƒ(x + nh) − nf {(x + (n − 1) h }
| + | n (n − 1) | ƒ {x + (n − 2) h } − ... + (−1)n ƒ(x) ]. |
| 2! |
(ii.)
As in the case of relation (i.) the limit expressed by the right-hand member may exist although some or all of the derived functions ƒ′(x), ƒ″(x), ... ƒ(n−1)(x) do not exist.
Corresponding to the rule iii. of § 11 we have the rule for forming the nth differential coefficient of a product in the form
| dn(uv) | = u | dnv | + n | du | dn−1v | + | n(n − 1) | d2u | dn−2v | + ... + | dnu | v, | |||
| dxn | dxn | dx | dxn−1 | 1·2 | dx2 | dxn−2 | dxn |