Editing Taylor's theorem (section)

=== Higher-order differentiability ===

A function {{math|''f'': '''R'''<sup>''n''</sup> → '''R'''}} is [[derivative|differentiable]] at {{math|'''''a''''' ∈ '''R'''<sup>''n''</sup>}} [[if and only if]] there exists a [[linear functional]] {{math|''L'' : '''R'''<sup>''n''</sup> → '''R'''}} and a function {{math|''h'' : '''R'''<sup>''n''</sup> → '''R'''}} such that

<math display="block"> f(\boldsymbol{x}) = f(\boldsymbol{a}) + L(\boldsymbol{x}-\boldsymbol{a}) + h(\boldsymbol{x})\lVert\boldsymbol{x}-\boldsymbol{a}\rVert,
\qquad \lim_{\boldsymbol{x}\to\boldsymbol{a}} h(\boldsymbol{x})=0. </math>

If this is the case, then <math display="inline">L = df(\boldsymbol{a})</math> is the (uniquely defined) [[differential of a function|differential]] of {{math|''f''}} at the point {{math|'''''a'''''}}. Furthermore, then the [[partial derivatives]] of {{math|''f''}} exist at {{math|'''''a'''''}} and the differential of {{math|''f''}} at {{math|'''''a'''''}} is given by

<math display="block"> df( \boldsymbol{a} )( \boldsymbol{v} ) = \frac{\partial f}{\partial x_1}(\boldsymbol{a}) v_1 + \cdots + \frac{\partial f}{\partial x_n}(\boldsymbol{a}) v_n. </math>

Introduce the [[multi-index notation]]

<math display="block"> |\alpha| = \alpha_1+\cdots+\alpha_n, \quad \alpha!=\alpha_1!\cdots\alpha_n!, \quad \boldsymbol{x}^\alpha=x_1^{\alpha_1}\cdots x_n^{\alpha_n} </math>

for {{math|''α'' ∈ '''N'''<sup>''n''</sup>}} and {{math|'''''x''''' ∈ '''R'''<sup>''n''</sup>}}.  If all the <math display="inline">k</math>-th order [[partial derivatives]] of {{math|''f'' : '''R'''<sup>''n''</sup> → '''R'''}} are continuous at {{math|'''''a''''' ∈ '''R'''<sup>''n''</sup>}}, then by [[symmetry of second derivatives|Clairaut's theorem]], one can change the order of mixed derivatives at {{math|'''''a'''''}}, so the short-hand notation

<math display="block"> D^\alpha f = \frac{\partial^{|\alpha|}f}{\partial\boldsymbol x^\alpha} = \frac{\partial^{\alpha_1 + \ldots + \alpha_n}f}{\partial x_1^{\alpha_1}\cdots \partial x_n^{\alpha_n}}</math>

for the higher order [[partial derivatives]] is justified in this situation. The same is true if all the ({{math|''k'' − 1}})-th order partial derivatives of {{math|''f''}} exist in some neighborhood of {{math|'''''a'''''}} and are differentiable at {{math|'''''a'''''}}.<ref>This follows from iterated application of the theorem that if the partial derivatives of a function {{math|''f''}} exist in a neighborhood of {{math|'''''a'''''}} and are continuous at {{math|'''''a'''''}}, then the function is differentiable at {{math|'''''a'''''}}.  See, for instance, {{harvnb|Apostol|1974|loc=Theorem 12.11}}.</ref> Then we say that {{math|''f''}} is {{math|''k''}} '''times differentiable at the point&nbsp;{{math|''a''}}'''.