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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_{h \to 0} \frac{f(a+h) - f(a)}{h}.

The lemma asserts that the existence of this derivative implies the existence of a function \varphi such that

:\lim_{h \to 0} \varphi(h) = 0 \qquad \text{and} \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+h) - f(a)}{h} - f'(a)

and verify this \varphi meets the requirements.

The lemma says, at least when h is sufficiently close to zero, that the difference quotient

:\frac{f(a+h) - f(a)}{h}

can be written as the derivative f' plus an error term \varphi(h) that vanishes at h=0.

That is, one has

:\frac{f(a+h) - f(a)}{h} = f'(a) + \varphi(h).

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{R}^n to \mathbb{R}. Then f is said to be differentiable at a if there is a linear function

:M: \mathbb{R}^n \to \mathbb{R}

and a function

:\Phi: D \to \mathbb{R}, \qquad D \subseteq \mathbb{R}^n \smallsetminus \{ \mathbf{0} \},

such that

:\lim_{\mathbf{h} \to 0} \Phi(\mathbf{h}) = 0 \qquad \text{and} \qquad f(\mathbf{a}+\mathbf{h}) - f(\mathbf{a}) = M(\mathbf{h}) + \Phi(\mathbf{h}) \cdot \Vert\mathbf{h}\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.

We can write the above equation in terms of the partial derivatives \frac{\partial f}{\partial x_i} as

: f(\mathbf{a} + \mathbf{h}) - f(\mathbf{a}) = \displaystyle\sum_{i=1}^n \frac{\partial f(a)}{\partial x_i} + \Phi(\mathbf{h}) \cdot \Vert\mathbf{h}\Vert

See also

References

  • {{cite web|url=http://clem.mscd.edu/~talmanl/PDFs/APCalculus/MultiVarDiff.pdf|title=Differentiability for Multivariable Functions|date=2007-09-12|accessdate=2012-06-28|first=Louis|last=Talman|archive-url=https://web.archive.org/web/20100620155743/http://clem.mscd.edu/~talmanl/PDFs/APCalculus/MultiVarDiff.pdf|archive-date=2010-06-20|url-status=dead}}
  • {{cite book|title=Calculus|first=James|last=Stewart|page=942|edition=7th|publisher=Cengage Learning|year=2008|isbn=978-0538498845}}
  • {{cite web|url=https://sites.math.washington.edu/~folland/Math134/lin-approx.pdf|title=Derivatives and Linear Approximation|first=Gerald|last=Folland}}

Category:Differential calculus