Fundamental increment lemma

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

$f'(a)=\lim _{h\to 0}{\frac {f(a+h)-f(a)}{h}}.$

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 ${\textstyle h}$. 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 $h=0$.

I.e. 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

• Generalizations of the derivative

References

• 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).