# Partial Derivative **The [[derivative]] of a multivariable function with respect to one of its variables whilst the others are held constant.** > [!Example] Partial Derivative > Let $f(x_{1},\dots,x_{i},\dots,x_{n})$ be a real-valued *multivariable* function defined on an *open subset* of $\mathbb{R}^{n}$ containing the point $(x_{1},\dots,x_{i},\dots,x_{n})$. > > The *partial derivative* of $f$ with respect to the $i$-th variable $x_{i}$ is defined as > $\frac{\partial}{\partial x_{i}}f(x_{1},\dots,x_{i},\dots,x_{n})=\lim_{h\rightarrow 0}\frac{f(x_{1},\dots,x_{i}+h,\dots,x_{n})-f(x_{1},\dots,x_{i},\dots,x_{n})}{h}$ Unlike with derivatives, the function may be *discontinuous* at a point even if *all* partial derivatives exist there. The process of determining a partial derivative is *partial differentiation*. >[!Explanation]+ > A partial derivative is defined as the derivative of a multivariable function with respect to one variable whilst *all others are held constant*. > > If all but one variable is held constant, the multivariable function becomes a *degenerate single variable function*. > > For example, in the diagram below, the graph of $f(x,y)=x^{2}+xy+y^{2}$ is shown intersecting with the plane $y=1$. > ![[PartialDerivativesExplanation1.svg]] > The points of intersection are shown with a red line, which represents $f(x,y)$ when $y$ is held at the arbitrary *constant* $1$. > > The multivariable function has become the *degenerate single variable function* $f(x,1)$, which is shown below in a simpler two-dimensional graph. > ![[PartialDerivativesExplanation2.svg]] > From the above diagram, it is clear that the [[Derivative#^1860be|same explanation]] relating the derivative to its limit definition can be applied here for the partial derivative. >[!Example using the limit definition]- > Suppose the partial derivative of $f(x,y)=x^{2}+xy+y^{2}$ is to be evaluated with respect to $x$ at $(2,1)$. > $\begin{align*} > \frac{\partial}{\partial x}f(x,y)&=\lim_{h\rightarrow 0}\frac{f(x+h,y)-f(x,y)}{h} \\ > &=\lim_{h\rightarrow 0}\frac{(x+h)^{2}+(x+h)y+y^{2}-(x^{2}+xy+y^{2})}{h} \\ > &=\lim_{h\rightarrow 0}\frac{x^{2}+2xh+h^{2}+xy+hy+y^{2}-x^{2}-xy-y^{2}}{h} \\ > &=\lim_{h\rightarrow 0}2x+h+y \\ > &=2x+y \\ > \therefore \frac{\partial}{\partial x}f(2,1)&=5 > \end{align*}$ >[!Example using the constant and constant factor rule]- > The partial differentiation of the previous example can be made simpler by using the [[Derivative#Constants|constant rule]] and the [[Derivative#^eed876|constant factor rule]]. > > For the previous example, by treating $y$ as a constant, > $\begin{align*} > f(x,y)&=x^{2}+xy+y^{2} \\ > \frac{\partial}{\partial x}f(x,y)&=2x+y \\ > \therefore \frac{\partial}{\partial x}f(2,1)&=5 > \end{align*}$ > which is the same result as obtained above. ## Notation ### Leibniz's notation Borrowing from Leibniz's notation for differentiation, the *first partial derivative* of the multivariable function $f(x,y,z)$ with respect to $x$ is denoted as $\frac{\partial f}{\partial x}, \frac{\partial}{\partial x}f$ The *$n$th partial derivative* with respect to $x$ is denoted as $\frac{\partial^{n}f}{\partial x^{n}}, \frac{\partial^{n}}{\partial x^{n}}f$ *Second order mixed derivatives* are denoted as $\frac{\partial^{2}f}{\partial x\partial y}, \frac{\partial^{2}}{\partial x\partial y}f$ >[!Explanation]- > The notation for mixed derivatives is an *abbreviation* of the repeated application of the *partial differential operator* with different variables of differentiation, for example: > $\frac{\partial^{2}f}{\partial x\partial y}=\frac{\partial}{\partial x}\left(\frac{\partial f}{\partial y}\right)$ *Higher order mixed derivatives* are denoted as $\frac{\partial^{i+j+k}f}{\partial x^{i}\partial y^{j}\partial z^{k}}, \frac{\partial^{i+j+k}}{\partial x^{i}\partial y^{j}\partial z^{k}}f$ ### Subscript notation Partial derivatives can be denoted with only the function symbol and the variable of differentiation in subscript. The *first and second partial derivatives* are denoted as $f_{x}, f_{xx}$ *Second order mixed derivatives* are denoted as $f_{yx}=\frac{\partial^{2}f}{\partial x\partial y}$ [^1] ### Euler's notation [[Derivative#Euler's notation|Euler's notation]] for partial derivatives simply replaces $D$ with $\partial$ to symbolise the *partial differential operator*. The *first partial derivative* with respect to $x$ is denoted as $\partial_{x}f=\frac{\partial f}{\partial x}$ The *second partial derivative* with respect to $x$ is denoted as $\partial_{xx}f=\partial^{2}_{x}f=\frac{\partial^{2} f}{\partial x^{2}}$ The *$n$th partial derivative* with respect to $x$ is denoted as $\partial_{x}^{n}f=\frac{\partial^{n}f}{\partial x^{n}}$ *Second order mixed derivatives* are denoted as $\partial_{yx}f=\frac{\partial^{2}f}{\partial x\partial y}$ [^1] [^1]: The subscript order of the variables is *reversed* compared to Leibniz's notation because it indicates the *order* in which the function was partially differentiated, i.e. with respect to $y$ then with respect to $x$. In Leibniz's notation, successive applications of the partial differential operator are written to the left. However, the order of mixed derivatives *does not change the result*.