The definition of derivative,
is not such a great way to actually find the derivative numerically. Its symmetric version,
performs much better in computations. For example, consider the derivative at the point . We know that . Numerically, with , we get
(error ) versus
Considering this, why don’t we ditch (1) altogether and adopt (2) as the definition of derivative? Just say that by definition,
whenever the limit exists.
This expands the class of differentiable functions: for example, becomes differentiable with . Which looks more like a feature than a bug: after all, has a minimum at , and the horizontal line through the minimum is the closest thing to the tangent line that it has.
Another example: the function
has under this definition, because
This example also makes sense: since , getting is expected. In fact, with the new definition we still have basic derivative rules: if are differentiable, then , , , are also differentiable (with the usual caveat about ) and the familiar formulas hold.
Since , the product on the right is . On the other hand,
which implies, by a computation similar to the above, that . So, if we want to have the chain rule (3), we must accept that
This is where the desire for high numerical precision leads.
Plenty of other things go wrong with the symmetric definition:
- Maximum or minimum of may be attained where exists and is nonzero.
- A differentiable function may be discontinuous.
- Having everywhere does not imply that is increasing.
- Mean Value Theorem fails.