Differentiability is the point at which multivariable calculus becomes linear algebra. A continuous map may carry nearby points to nearby points, but a differentiable map does something more rigid: near a chosen point, it is well approximated by a single [linear map](/page/Linear%20Map). This local linear model is what makes the chain rule possible, turns nonlinear equations into tangent-level questions, and connects [continuity](/page/Continuity), [partial derivatives](/page/Partial%20Derivative), the [Jacobian matrix](/page/Jacobian%20Matrix), and the [inverse function theorem](/theorems/51).

The definition is stronger than having directional or partial derivatives. It asks for one linear approximation that works uniformly for all small displacement vectors. That single map is the total derivative, and it is the object that survives under coordinate changes, composition, optimization, and geometric constructions.

[motivation] In one-variable calculus, differentiability at $a$ means that the graph has a best affine approximation near $a$: \begin{align*} f(a+h) = f(a) + f'(a)h + o(|h|). \end{align*} For maps $f: U \subset \mathbb{R}^m \to \mathbb{R}^n$, the number $f'(a)$ must be replaced by a linear map from input displacements to output displacements. The definition of differentiability is designed to isolate exactly this first-order behaviour. The key point is that the error must be small compared with $|h|$, not merely small in absolute size. If the error is only known to vanish, the map is continuous. If the error vanishes after division by $|h|$, the linear part captures the whole first-order motion of the map. [/motivation]

## Definition

The central question is whether the increment $f(a+h)-f(a)$ has a linear part that explains it to first order. Since $h$ ranges through vectors in $\mathbb{R}^m$, the candidate first-order model must be a linear map on $\mathbb{R}^m$ with values in $\mathbb{R}^n$.

[definition: Differentiable Map] Let $U \subset \mathbb{R}^m$ be open, let $a \in U$, and let $f: U \to \mathbb{R}^n$ be a function. The map $f$ is differentiable at $a$ if there exist a linear map $L \in \mathcal{L}(\mathbb{R}^m, \mathbb{R}^n)$ and a function $r: U-a \to \mathbb{R}^n$ such that $r(0)=0$, $r$ is continuous at $0$, and for every $h \in \mathbb{R}^m$ with $a+h \in U$, \begin{align*} f(a+h) = f(a) + L(h) + |h|r(h). \end{align*} [/definition]

A reusable calculus needs a name for the unique first-order model at a point. Naming that linear map allows later statements to refer to the first-order approximation without repeating the full remainder condition.

[definition: Total Derivative] Let $U \subset \mathbb{R}^m$ be open, let $a \in U$, and let $f: U \to \mathbb{R}^n$ be differentiable at $a$. The total derivative of $f$ at $a$ is the unique linear map $Df_a: \mathbb{R}^m \to \mathbb{R}^n$ such that \begin{align*} f(a+h)=f(a)+Df_a(h)+o(|h|) \end{align*} as $h \to 0$ with $a+h \in U$. [/definition]

A single point condition is not enough for calculus rules that must hold uniformly as the base point moves. Since compositions, derivative maps, and coordinate formulas all require differentiability at each point where they are applied, we need a domain-level condition rather than repeated pointwise hypotheses.

[definition: Differentiable Map on an Open Set] Let $U \subset \mathbb{R}^m$ be open and let $f: U \to \mathbb{R}^n$ be a function. The map $f$ is differentiable on $U$ if $f$ is differentiable at every point $a \in U$. [/definition]

A theory of regularity also needs to compare derivatives at different points. Treating $a\mapsto Df_a$ as a function is the gateway to continuous differentiability, higher derivatives, and estimates that vary across the domain.

[definition: Derivative Map] Let $U \subset \mathbb{R}^m$ be open and let $f: U \to \mathbb{R}^n$ be differentiable on $U$. The derivative map of $f$ is the function $Df: U \to \mathcal{L}(\mathbb{R}^m, \mathbb{R}^n)$ defined by $a \mapsto Df_a$. [/definition]

Concrete calculations often require coordinates. In Euclidean space the standard bases convert each total derivative into a rectangular array of partial derivatives, which gives the computational object used in change-of-variables and linearization formulas.

[definition: Jacobian Matrix] Let $U \subset \mathbb{R}^m$ be open, let $a \in U$, and let $f=(f_1,\ldots,f_n):U\to\mathbb{R}^n$ be differentiable at $a$. The Jacobian matrix of $f$ at $a$ is the matrix $Jf_a \in \mathbb{R}^{n\times m}$ whose entries are \begin{align*} (Jf_a)_{ij}=\partial_{x_j}f_i(a), \qquad 1\le i\le n,\quad 1\le j\le m. \end{align*} [/definition]

The Jacobian matrix acts on column vectors by matrix multiplication and represents the same first-order map: $Df_a(h)=Jf_a h$. This distinction matters in geometry and functional analysis, where the derivative is intrinsically a linear map and a matrix only after bases have been chosen.

## Equivalent Characterisations

The remainder term definition is precise, but in estimates it is often more convenient to divide by $|h|$ and express the condition as a limiting statement. This formulation makes the phrase "first-order approximation" literal and provides the standard test used in computations.

[quotetheorem:319]

This limit form is often the practical way to verify differentiability: after proposing a candidate linear map, one estimates the error divided by $|h|$ and checks that it tends to zero. It also clarifies the limitation of first-order calculus: differentiability controls only the leading linear part, while all higher-order behaviour is hidden in the vanishing remainder.

Coordinate partial derivatives are often easier to compute than the total derivative, but by themselves they do not automatically control the full multivariable remainder. The key obstruction is whether the coordinate data vary regularly enough near the point to assemble into one linear approximation.