Fundamental increment lemma

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 dimensions

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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 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   as

 

See also

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References

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  • Talman, Louis (2007-09-12). "Differentiability for Multivariable Functions" (PDF). Archived from the original (PDF) on 2010-06-20. Retrieved 2012-06-28.
  • Stewart, James (2008). Calculus (7th ed.). Cengage Learning. p. 942. ISBN 978-0538498845.
  • Folland, Gerald. "Derivatives and Linear Approximation" (PDF).