Fundamental increment lemma

In that the existence of ${\displaystyle \varphi }$ uniquely characterises the number ${\displaystyle f'(a)}$, 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 ${\displaystyle \mathbb {R} ^{n}}$ to ${\displaystyle \mathbb {R} }$. Then f is said to be differentiable at a if there is a linear function

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

and a function

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

such that

${\displaystyle \lim _{\mathbf {h} \to 0}\Phi (\mathbf {h} )=0\qquad {\text{and}}\qquad f(\mathbf {a} +\mathbf {h} )-f(\mathbf {a} )=M(\mathbf {h} )+\Phi (\mathbf {h} )\cdot \Vert \mathbf {h} \Vert }$

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 ${\displaystyle {\frac {\partial f}{\partial x_{i}}}}$ as

${\displaystyle f(\mathbf {a} +\mathbf {h} )-f(\mathbf {a} )=\displaystyle \sum _{i=1}^{n}{\frac {\partial f(a)}{\partial x_{i}}}h_{i}+\Phi (\mathbf {h} )\cdot \Vert \mathbf {h} \Vert }$

• Generalizations of the derivative

• 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).