In single-variable differential calculus, the fundamental increment lemma is an immediate consequence of the definition of the derivative of a function at a point :
The lemma asserts that the existence of this derivative implies the existence of a function such that
for sufficiently small but non-zero . For a proof, it suffices to define
and verify this meets the requirements.
The lemma says, at least when is sufficiently close to zero, that the difference quotient
can be written as the derivative f' plus an error term that vanishes at .
I.e. one has,
Differentiability in higher dimensionsedit
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
Talman, Louis (2007-09-12). (PDF). Archived from the original (PDF) on 2010-06-20. Retrieved 2012-06-28. {{cite web}}: More than one of |archivedate= and |archive-date= specified (help); More than one of |archiveurl= and |archive-url= specified (help)
Stewart, James (2008). Calculus (7th ed.). Cengage Learning. p. 942. ISBN978-0538498845.
Folland, Gerald. "Derivatives and Linear Approximation" (PDF).
January 01, 1970
fundamental, increment, lemma, single, variable, differential, calculus, fundamental, increment, lemma, immediate, consequence, definition, derivative, textstyle, function, textstyle, point, textstyle, displaystyle, frac, lemma, asserts, that, existence, this,. In single variable differential calculus the fundamental increment lemma is an immediate consequence of the definition of the derivative f a textstyle f a of a function f textstyle f at a point a textstyle a f a lim h 0 f a h f a h displaystyle 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 f displaystyle varphi such that lim h 0 f h 0 and f a h f a f a h f h h displaystyle 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 textstyle h For a proof it suffices to define f h f a h f a h f a displaystyle varphi h frac f a h f a h f a and verify this f displaystyle varphi meets the requirements The lemma says at least when h displaystyle h is sufficiently close to zero that the difference quotient f a h f a h displaystyle frac f a h f a h can be written as the derivative f plus an error term f h displaystyle varphi h that vanishes at h 0 displaystyle h 0 I e one has f a h f a h f a f h displaystyle frac f a h f a h f a varphi h Differentiability in higher dimensions editIn that the existence of f displaystyle varphi nbsp uniquely characterises the number f a displaystyle f a nbsp 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 R n displaystyle mathbb R n nbsp to R displaystyle mathbb R nbsp Then f is said to be differentiable at a if there is a linear function M R n R displaystyle M mathbb R n to mathbb R nbsp and a function F D R D R n 0 displaystyle Phi D to mathbb R qquad D subseteq mathbb R n smallsetminus mathbf 0 nbsp such that lim h 0 F h 0 and f a h f a M h F h h displaystyle 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 nbsp 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 f x i displaystyle frac partial f partial x i nbsp as f a h f a i 1 n f a x i F h h displaystyle 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 nbsp See also editGeneralizations of the derivativeReferences editTalman Louis 2007 09 12 Differentiability for Multivariable Functions PDF Archived from the original PDF on 2010 06 20 Retrieved 2012 06 28 a href Template Cite web html title Template Cite web cite web a More than one of archivedate and archive date specified help More than one of archiveurl and archive url specified help Stewart James 2008 Calculus 7th ed Cengage Learning p 942 ISBN 978 0538498845 Folland Gerald Derivatives and Linear Approximation PDF Retrieved from https en wikipedia org w index php title Fundamental increment lemma amp oldid 1223849467, wikipedia, wiki, book, books, library,