Let be open,
is an arbitrary point, and
is a function. Recall that
is called differentiable at
if there is a linear map, denoted by
, such that
The linear map is called derivative of
at
, denoted by
. Notice that in the nominator of the above quotient, the symbol
is simply the absolute value function. But this is no longer true for higher-order derivatives that we are going to define.
The following theorem is well-known.
Theorem 1 (1st order differentiability). The function
is differentiable at
if all partial derivatives
exist in a neighborhood of
and are continuous at
.
When exists, we must have
In this note, we want to extend the above theorem for higher-order derivatives. To be more precise, we prove the following
(more…)Theorem 2 (2nd order differentiability). The function
is twice differentiable at
if all partial derivatives
exist in a neighborhood of
and are continuous at
.