A function of several variables can fail to have an ordinary one-dimensional derivative for the simple reason that there is no single direction in which to move. At a point $a \in \mathbb{R}^m$, the expression $f(a+h)-f(a)$ depends on the whole vector $h$, and different approaches to $a$ can see different behaviour. Partial derivatives isolate the most basic probes: move parallel to one coordinate axis, freeze all other variables, and measure the resulting one-dimensional rate of change.

This coordinate-axis viewpoint is deliberately modest. It does not try to describe the full [derivative](/page/Derivative) at once. Instead, it asks for the response of the function to each coordinate input separately. That makes partial derivatives computationally accessible and central in calculus, differential equations, optimization, complex analysis, and differential geometry.

The price of this accessibility is that partial derivatives can mislead. A function may have every partial derivative at a point and still fail to be continuous there. Even when all first partial derivatives exist, they may not combine into a linear approximation. The chapter therefore treats partial derivatives as local coordinate tests: powerful, necessary for differentiability, but not by themselves the same as differentiability.

[example: Axis Tests Can Miss Two-Dimensional Behaviour] Let $f:\mathbb{R}^2\to\mathbb{R}$ satisfy $f(0,0)=0$ and, for $(x,y)\ne(0,0)$, \begin{align*} f(x,y)=\frac{xy}{x^2+y^2}. \end{align*} We compute the two coordinate-axis difference quotients at the origin. For $t\ne0$, \begin{align*} f(t,0)=\frac{t\cdot 0}{t^2+0^2}=0. \end{align*} Hence \begin{align*} \frac{f(t,0)-f(0,0)}{t}=\frac{0-0}{t}=0. \end{align*} Taking $t\to0$ gives $\partial_x f(0,0)=0$. Similarly, for $t\ne0$, \begin{align*} f(0,t)=\frac{0\cdot t}{0^2+t^2}=0. \end{align*} Therefore \begin{align*} \frac{f(0,t)-f(0,0)}{t}=\frac{0-0}{t}=0, \end{align*} so $\partial_y f(0,0)=0$. The diagonal approach gives different behaviour. For $x\ne0$, \begin{align*} f(x,x)=\frac{x\cdot x}{x^2+x^2}. \end{align*} Since $x^2+x^2=2x^2$ and $x^2\ne0$, \begin{align*} f(x,x)=\frac{x^2}{2x^2}=\frac{1}{2}. \end{align*} Thus along the path $(x,x)\to(0,0)$ the function values stay equal to $1/2$, while $f(0,0)=0$. The coordinate-axis partial derivatives both exist and vanish, but the function is not continuous at the origin. [/example]

This example gives the main warning for the chapter. Partial derivatives are not a replacement for the total derivative; they are the coordinate pieces from which differentiability may sometimes be reconstructed, provided additional regularity is present.

## Definition

The most direct way to define a partial derivative is to reduce the multivariable function to a one-variable function by holding all but one coordinate fixed. The openness of the domain matters because we need to move slightly in both positive and negative coordinate directions while remaining inside the domain.

[definition: Partial Derivative] Let $U \subset \mathbb{R}^m$ be open, let $f: U \to \mathbb{R}^n$ be a function, let $a=(a_1,\ldots,a_m) \in U$, and let $i \in \{1,\ldots,m\}$. The $i$-th partial derivative of $f$ at $a$ is \begin{align*} \partial_{x_i} f(a) := \lim_{t \to 0} \frac{f(a+t e_i)-f(a)}{t}, \end{align*} provided this limit exists in $\mathbb{R}^n$, where $e_i$ is the $i$-th standard basis vector of $\mathbb{R}^m$. [/definition]

The notation emphasizes the coordinate $x_i$ rather than an abstract direction. In the definition-family graph, this page is a child of [Derivative](/page/Derivative): the partial derivative is the [directional derivative](/page/Directional%20Derivative) in the coordinate direction $e_i$, while the total derivative, when it exists, is a [linear map](/page/Linear%20Map) controlling all small vectors $h \in \mathbb{R}^m$ at once.

[remark: Scalar and Vector-Valued Cases] If $n=1$, then $\partial_{x_i} f(a)$ is a real number. If $f=(f_1,\ldots,f_n):U\to\mathbb{R}^n$, then \begin{align*} \partial_{x_i} f(a)=\bigl(\partial_{x_i} f_1(a),\ldots,\partial_{x_i} f_n(a)\bigr) \end{align*} whenever all component partial derivatives exist. [/remark]

For functions on the plane, the notation often records the visible coordinate names. This is the form used in elementary calculus, and it is the same definition with $m=2$.

[example: Partial Derivatives in Two Variables] Let $U=\mathbb{R}^2$ and let $f:U\to\mathbb{R}$ be given by \begin{align*} f(x,y)=x^2y+\sin(xy). \end{align*} Fix $(x,y)\in\mathbb{R}^2$. For the $x$-partial derivative, the difference quotient is \begin{align*} \frac{f(x+t,y)-f(x,y)}{t}=\frac{(x+t)^2y+\sin((x+t)y)-x^2y-\sin(xy)}{t}. \end{align*} Expanding $(x+t)^2=x^2+2xt+t^2$ gives \begin{align*} \frac{(x+t)^2y-x^2y}{t}=\frac{(x^2+2xt+t^2)y-x^2y}{t}. \end{align*} The numerator is $x^2y+2xty+t^2y-x^2y=2xty+t^2y$, so \begin{align*} \frac{(x+t)^2y-x^2y}{t}=2xy+ty. \end{align*} For the sine term, write $(x+t)y=xy+ty$. Since $\frac{d}{ds}\sin s=\cos s$ and $s(t)=xy+ty$ has derivative $y$, the one-variable chain rule gives \begin{align*} \lim_{t\to0}\frac{\sin((x+t)y)-\sin(xy)}{t}=y\cos(xy). \end{align*} Therefore \begin{align*} \partial_x f(x,y)=\lim_{t\to0}\left(2xy+ty+\frac{\sin((x+t)y)-\sin(xy)}{t}\right)=2xy+y\cos(xy). \end{align*} For the $y$-partial derivative, the difference quotient is \begin{align*} \frac{f(x,y+t)-f(x,y)}{t}=\frac{x^2(y+t)+\sin(x(y+t))-x^2y-\sin(xy)}{t}. \end{align*} The polynomial part satisfies \begin{align*} \frac{x^2(y+t)-x^2y}{t}=\frac{x^2y+x^2t-x^2y}{t}=x^2. \end{align*} For the sine term, write $x(y+t)=xy+xt$. Since $\frac{d}{ds}\sin s=\cos s$ and $s(t)=xy+xt$ has derivative $x$, the one-variable chain rule gives \begin{align*} \lim_{t\to0}\frac{\sin(x(y+t))-\sin(xy)}{t}=x\cos(xy). \end{align*} Thus \begin{align*} \partial_y f(x,y)=x^2+x\cos(xy). \end{align*} The two formulas are different because the first coordinate probe changes $x$ while keeping $y$ fixed, and the second changes $y$ while keeping $x$ fixed. [/example]

A partial derivative at a single point is useful, but analysis often needs the derivative as a function of the point. That leads to the partial derivative function on the subset where the pointwise limits exist.

[definition: Partial Derivative Function] Let $U \subset \mathbb{R}^m$ be open and let $f:U\to\mathbb{R}^n$ be a function. If $\partial_{x_i}f(a)$ exists for every $a\in U$, the $i$-th partial derivative function is the map $\partial_{x_i}f:U\to\mathbb{R}^n$ given by $a\mapsto \partial_{x_i}f(a)$. [/definition]

Once partial derivatives are functions, we can ask whether they are continuous, integrable, bounded, or differentiable again. These extra properties are what allow coordinate calculations to support theorems about full differentiability, Taylor expansions, and PDE.

## Coordinate Directions and the Total Derivative

### Jacobian Data

The total derivative asks for a linear map that approximates every small displacement. Partial derivatives ask only what happens on the coordinate axes. The relationship is essential: differentiability forces the partial derivatives to exist, and the partial derivatives are the columns of the Jacobian matrix.

Before connecting axis derivatives to full differentiability, we need a single object that stores every coordinate response. The Jacobian matrix is the bookkeeping device that makes the possible linear approximation visible.

[definition: Jacobian Matrix] Let $U\subset\mathbb{R}^m$ be open, let $f=(f_1,\ldots,f_n):U\to\mathbb{R}^n$, and let $a\in U$. If each partial derivative $\partial_{x_j}f_i(a)$ exists, the Jacobian matrix of $f$ at $a$ is the matrix $Jf_a\in\mathbb{R}^{n\times m}$ with entries \begin{align*} (Jf_a)_{ij}=\partial_{x_j}f_i(a). \end{align*} [/definition]