Fundamental increment lemma explained

In single-variable differential calculus, the fundamental increment lemma is an immediate consequence of the definition of the derivative $f\text{'}(a)$ of a function $f$ at a point $a$:

f'(a) = \lim _{h} (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} \varphi(h) = 0 and 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) = (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(f(a + h) - f(a))/(h) can be written as the derivative f plus an error term \varphi(h) that vanishes at h =0 .

I.e. one has,

(f(a + h) - f(a))/(h) = f'(a) + \varphi(h).

Differentiability in higher dimensions

In that the existence of

M: R ^{n} \to R and a function \Phi: D \to R , D \subseteq R ^{n} \smallsetminus \{ 0 \}, such that \lim _{h} \Phi( h ) = 0 and f( a + h ) - f( a ) = M( h ) + \Phi( h ) ⋅ \Vert 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

(\partial f)/(\partial x _{i}) asf( a + h ) - f( a ) = \displaystyle\sum ^{n} _{i =1} (\partial f(a))/(\partial x _{i}) + \Phi( h ) ⋅ \Vert h \Vert

See also

• Generalizations of the derivative

References

• Web site: Differentiability for Multivariable Functions. 2007-09-12. 2012-06-28. Louis. Talman. https://web.archive.org/web/20100620155743/http://clem.mscd.edu/~talmanl/PDFs/APCalculus/MultiVarDiff.pdf. 2010-06-20. dead.
• Book: Stewart, James. Calculus. 942. 7th. Cengage Learning. 2008. 978-0538498845.
• Web site: Derivatives and Linear Approximation. Gerald. Folland.