fundamental increment lemma

In single-variable differential calculus, the fundamental increment lemma is an immediate consequence of the definition of the derivative ''f''(''a'') of a function ''f'' at a point ''a'':
:$f'(a) = \lim_ \frac.$
The lemma asserts that the existence of this derivative implies the existence of a function $\varphi$ such that
:$\lim_ \varphi(h) = 0 \qquad \text \qquad f(a+h) = f(a) + f'(a)h + \varphi(h)h$
for sufficiently small but non-zero ''h''. For a proof, it suffices to define
:$\varphi(h) = \frac - f'(a)$
and verify this $\varphi$ meets the requirements.

Differentiability in higher dimensions

In that the existence of $\varphi$ uniquely characterises the number $f'(a)$, 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 $\mathbb^n$ to $\mathbb$. Then ''f'' is said to be differentiable at a if there is a linear function
:$M: \mathbb^n \to \mathbb$
and a function
:$\Phi: D \to \mathbb, \qquad D \subseteq \mathbb^n \smallsetminus \,$
such that
:$\lim_ \Phi(\mathbf) = 0 \qquad \text \qquad f(\mathbf+\mathbf) = f(\mathbf) + M(\mathbf) + \Phi(\mathbf) \cdot \Vert\mathbf\Vert$
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 derivatives) of ''f'' at a. Notably, ''M'' is given by the Jacobian matrix 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