Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/215

123. Partial derivatives. Since $x$ and $y$ are independent in

$z=f(x,\ y),$

$x$ may be supposed to vary while $y$ remains constant, or the reverse.

The derivative of $z$ with respect to $x$ when $x$ varies and $y$ remains constant^[1] is called the partial derivative of $z$ with respect to $x$, and is denoted by the symbol ${\tfrac {\partial z}{\partial z}}$ We may then write

(A)	${\frac {\partial z}{\partial x}}=\lim _{\Delta x\to 0}\left[{\frac {f(x+\Delta x,y)-f(x,y)}{\Delta x}}\right]$

Similarly, when $x$ remains constant^[1] and $y$ varies, the partial derivative of z with respect to $y$ is

(B)	${\frac {\partial z}{\partial y}}=\lim _{\Delta y\to 0}\left[{\frac {f(x,y+\Delta y)-f(x,y)}{\Delta y}}\right]$
	${\frac {\partial z}{\partial y}}$ is also written ${\frac {\partial }{\partial x}}f(x,y),$ or ${\frac {\partial f}{\partial x}}.$
Similarly,	${\frac {\partial z}{\partial y}}$ is also written ${\frac {\partial }{\partial y}}f(x,y),$ or ${\frac {\partial f}{\partial y}}$.

In order to avoid confusion the round $\partial$^[2] has been generally adopted to indicate partial differentiation. Other notations; however, which are in use are

$\left({\frac {dz}{dx}}\right),\left({\frac {dz}{dy}}\right);\ f'_{x}(x,y),f'_{y}(x,y);\ f_{x}(x,y),f_{y}(x,y);\ D_{x}f,D_{y}f;z_{x},z_{y}.$

Our notation may be extended to a function of any number of independent variables. Thus, if

$u=F(x,\ y,\ z),$

then we have the three partial derivatives

${\frac {\partial u}{\partial x}},{\frac {\partial u}{\partial y}},{\frac {\partial u}{\partial z}}$; or, ${\frac {\partial F}{\partial x}},{\frac {\partial F}{\partial y}},{\frac {\partial F}{\partial z}}.$

Illustrative Example 1. Find the partial derivatives of $z=ax^{2}+2bxy+cy^{2}.$

Solution.	${\tfrac {\partial z}{\partial x}}=2ax+2by$, treating $y$ as a constant,
	${\tfrac {\partial z}{\partial y}}=2bx+cy$, treating $x$ as a constant.

1. The constant values are substituted in the function before differentiating
2. Introduced by [[1]] (1804-1851)