Fundamental increment lemma

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

$f^{\prime }(a)=\underset{h\to 0}{\lim }\frac{f(a+h)-f(a)}{h}.$

The lemma asserts that the existence of this derivative implies the existence of a function $\phi$ such that

$\underset{h\to 0}{\lim }\phi (h)=0\text{and}f(a+h)=f(a)+f^{\prime }(a)h+\phi (h)h$

for sufficiently small but non-zero $h$. For a proof, it suffices to define

$\phi (h)=\frac{f(a+h)-f(a)}{h}-f^{\prime }(a)$

and verify this $\phi$ 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 $\phi (h)$ that vanishes at $h=0$.

I.e. one has,

$\frac{f(a+h)-f(a)}{h}=f^{\prime }(a)+\phi (h).$

Differentiability in higher dimensions

In that the existence of $\phi$ uniquely characterises the number $f^{\prime }(a)$, 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

$\mathrm{\Phi }:D\to \mathbb{ℝ},D\subseteq {\mathbb{ℝ}}^n\setminus \{\mathbf{𝟎}\},$

such that

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

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

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

See also

• Generalizations of the derivative

References

• Talman, Louis (2007-09-12). "Differentiability for Multivariable Functions". http://clem.mscd.edu/~talmanl/PDFs/APCalculus/MultiVarDiff.pdf. Retrieved 2012-06-28. 
• Stewart, James (2008). Calculus (7th ed.). Cengage Learning. p. 942. ISBN 978-0538498845. 
• Folland, Gerald. "Derivatives and Linear Approximation". https://sites.math.washington.edu/~folland/Math134/lin-approx.pdf.