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

(error ).

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

and

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.

Let’s test the chain rule on the function . The rule says

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.

Are you trying to imply that a desire for high numerical precision is bad?

I suppose it’s just the irony of the situation 🙂

I do like precision-improving tricks, such as https://calculus7.org/2013/06/22/improving-the-wallis-product/

But I also like to throw in a polemical statement now and then: https://calculus7.org/2012/06/30/the-definition-of-uniform-continuity-is-wrong/

But for large values of 2 and 5, right?