Total Derivative as a Linear Map¶

Definition¶

Let $E\subseteq \mathbb{R}^n$ $R^{n}$ , let $\boldsymbol{f}\colon E\to\mathbb{R}^k$ be a function, let $\boldsymbol{x}_0\in\operatorname{int}E$ $int E$ , and let $\boldsymbol{L}\colon \mathbb{R}^n\to\mathbb{R}^k$ be a linear transformation. We say that $\boldsymbol{f}$ is (totally) differentiable at $\boldsymbol{x}=\boldsymbol{x}_{0}$ with (total) derivative $\boldsymbol{L}$ if, as $\boldsymbol{x}\to\boldsymbol{x}_0$ $x_{0}$ ,

\frac{\|\boldsymbol{f}(\boldsymbol{x})-\boldsymbol{f}(\boldsymbol{x}_0)-\boldsymbol{L}(\boldsymbol{x}-\boldsymbol{x}_0)\|}{\|\boldsymbol{x}-\boldsymbol{x}_0\|}\to0.

The limit must hold however $\boldsymbol{x}$ approaches $\boldsymbol{x}_0$ , that is, along every convergent sequence $\boldsymbol{x}_j\to\boldsymbol{x}_0$ with $\boldsymbol{x}_j\neq\boldsymbol{x}_0$ $x_{0}$ , not just along coordinate lines. It says that the relative error in the linear approximation tends to zero.

Interpretation¶

Total differentiability means the function can be approximated near the point by a linear transformation, with the error shrinking to zero faster than the distance to the point. The linear map $\boldsymbol{h}\mapsto\boldsymbol{L}\boldsymbol{h}$ is also called the differential of $\boldsymbol{f}$ at $\boldsymbol{x}_0$ ; for $n=k=1$ it is $h\mapsto f'(x_0)h$ $f^{'} (x_{0}) h$ , the one-variable differential $\mathrm{d}y=f'(x)\,\mathrm{d}x$ $f^{'} (x) d x$ .

It is the multidimensional version of the one-variable fact that $f$ is differentiable at $x_0$ if and only if $f(x_0+h)=f(x_0)+f'(x_0)h+o(h)$ as $h\to0$ (sometimes called Newton's approximation).

A small grid around x0 with a vector h, mapped by f onto a slanted grid around f(x0) with the image vector Lh: to first order, small squares become small parallelograms.

The first-order approximation is $\boldsymbol{f}(\boldsymbol{x}_0+\boldsymbol{h})=\boldsymbol{f}(\boldsymbol{x}_0)+\boldsymbol{L}\boldsymbol{h}+o(\|\boldsymbol{h}\|)$ $o (∥ h ∥)$ , where $\boldsymbol{L}=\boldsymbol{Df}(\boldsymbol{x}_0)$ $Df (x_{0})$ . Locally, $\boldsymbol{f}$ acts like this linear map, so small squares become approximate parallelograms.

Move $\boldsymbol x_0$ and zoom in: near $\boldsymbol x_0$ the graph is almost the plane $f(\boldsymbol x_0)+\mathrm{D}f_{\boldsymbol x_0}(\boldsymbol h)$ $D f_{x_{0}} (h)$ , with an error that shrinks faster than $\|\boldsymbol h\|$ .

Key Results¶

For the function $\boldsymbol{f}\colon \mathbb{R}^n\to\mathbb{R}^k$ at $\boldsymbol{x}=\boldsymbol{x}_0$ $x_{0}$ , its derivative, $\mathrm{D} \boldsymbol{f}_{\boldsymbol{x}_0}$ , is a $k\times n$ matrix or an equivalent linear transformation from $\mathbb{R}^n$ to $\mathbb{R}^k$ .
$\mathrm{D} \boldsymbol{f}_{\boldsymbol{x}_0}(\boldsymbol{h})$ , where $\boldsymbol{h}$ is an infinitesimal vector, provides an approximation of the shift in $\boldsymbol{f}$ (a vector in $\mathbb{R}^k$ ) arising from the change $\boldsymbol{h}$ in $\boldsymbol{x}$ around $\boldsymbol{x}=\boldsymbol{x}_0$ (a vector in $\mathbb{R}^n$ ).

Examples¶

The function $\boldsymbol{f}(x,y) = (x^2,y^2)$ is differentiable at $\boldsymbol{x}=(1,2)$ with the derivative $\boldsymbol{L}(x,y) = (2x,4y)$ $(2 x, 4 y)$ .
The function $f(x,y) = \sin(x^2+y)$ is differentiable at $(x,y)=(0,0)$ and the derivative is $L(x,y)=y$ $y$ .

Prerequisites¶

Sources¶

Math III: Analysis · Lecture 11

Aliases: total derivative, Frechet derivative, Fréchet derivative, totally differentiable Keywords: differential Symbols: element of, real numbers, subset

Last updated: 17 October 2026 21:53