# Definitions > [!NOTE] Definition (General) > Let $n\in \mathbb{N}^{+}$ be a [[Natural Numbers|positive natural number]]. Let $\mathbb{R}^{n}$ denote the [[Real n-Space|real n-space]]. Let $U\subset \mathbb{R}^{n}$ be a [[Subset|subset]]. Let $f:U\to \mathbb{R}$ be a [[Real-Valued Function on Real n-Space (Multivariable Function)|multivariable function]]. Let $\underline{a}=(a_{1},a_{2},a_{3},\dots,a_{n})\in \mathbb{R}^{n}.$ For $i=1,2,\dots,n,$ the $i$th partial derivative of $f$ at $\underline{a}$ is defined as the [[Limit of Real Function at a Point|limit]]: $\frac{ \partial }{ \partial x_{i} } f\bigg |_{\underline{a}} = \lim_{ x_{i} \to a_{i} } \frac{f(a_{1},a_{2},\dots,x_{i},\dots,a_{n})-f(\underline{a})}{x_{i}-a_{i}} =\lim_{ h \to 0 } \frac{f(\underline{a}+h\underline{e}_{i}) -f(\underline{a})}{h}$that is, the [[Derivative of Real Function|derivative]] of the real function $g_{i}$ defined by $g_{i}(x)=f(a_{1},a_{2},\dots,x,\dots a_{n})$ in the case that it exists. > > In other words, the $i$th partial derivative is the directional derivative in the direction of the $i$th standard basis of $\mathbb{R}^{n}.$ **Notation**: If $f:U\subset \mathbb{R}^{3} \to \mathbb{R},$ we may write $\frac{ \partial f }{ \partial x },\frac{ \partial f }{ \partial y },\frac{ \partial f }{ \partial z }$ for its first, second and third partial derivatives. # Properties If all mixed second order partial derivatives are continuous at a point (or on a set), the partial derivatives can be exchanged by [[Criterion for Equality of Mixed Partial Derivatives (Clairaut's Theorem)]]: $\frac{ \partial^{2} f }{ \partial x \partial y } = \frac{ \partial^{2} f }{ \partial y \partial x } .$ By [[Chain Rule for Differentiability]]: for $g(x,y,z)$ where $x,y,z$ are themselves functions of $t$, $\frac{dg}{dt} = \frac{ \partial g }{ \partial x } \frac{dx}{dt} + \frac{ \partial g }{ \partial y } \frac{dy}{dt} + \frac{ \partial g }{ \partial z } \frac{dz}{dt}.$ The [[Partial Derivatives of Product of Real-Valued Functions of Several Real Variables (Product Rule)|product rule]] certainly follows trivially. Let $f,g$ be functions on $(x,y,z)$ then $\frac{ \partial }{ \partial x }(fg) = g \frac{ \partial f }{ \partial x } +f \frac{ \partial g }{ \partial x } .$ # Applications Examples > [!Example] > Given $f(x,y)=10x-x^{2}y$ and point $P(1,1)$. > 1. Evaluate $f$ at $P$ > 2. Evaluate the partial derivatives of $f$ at $P$. > >Solution >1. $f(1,1)=10-1=9$ >2. $\frac{ \partial f }{ \partial x }=10-2xy$; $\frac{ \partial f }{ \partial x }(1,1)=8$. Similarly $\frac{ \partial f }{ \partial y }=-x^{2}$; $\frac{ \partial f }{ \partial y }=-1$. - [[Gradient of Real-Valued Function on Real n-Space]].