In single-variable
differential calculus
In mathematics, differential calculus is a subfield of calculus that studies the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus—the study of the area beneath a curve. ...
, the fundamental increment lemma is an immediate consequence of the definition of the
derivative ''f''(''a'') of a
function ''f'' at a point ''a'':
:
The lemma asserts that the existence of this derivative implies the existence of a function
such that
:
for sufficiently small but non-zero ''h''. For a proof, it suffices to define
:
and verify this
meets the requirements.
Differentiability in higher dimensions
In that the existence of
uniquely characterises the number
, the fundamental increment lemma can be said to characterise the
differentiability of single-variable functions. For this reason, a generalisation of the lemma can be used in the definition of differentiability in
multivariable calculus. In particular, suppose ''f'' maps some subset of
to
. Then ''f'' is said to be differentiable at a if there is a
linear function
:
and a function
:
such that
:
for non-zero h sufficiently close to 0. In this case, ''M'' is the unique derivative (or
total derivative, to distinguish from the
directional and
partial derivative
In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Part ...
s) of ''f'' at a. Notably, ''M'' is given by the
Jacobian matrix
In vector calculus, the Jacobian matrix (, ) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. When this matrix is square, that is, when the function takes the same number of variables as ...
of ''f'' evaluated at a.
See also
*
Generalizations of the derivative
References
*
*{{cite book, title=Calculus, first=James, last=Stewart, page=942, edition=7th, publisher=Cengage Learning, year=2008, isbn=978-0538498845
Differential calculus