Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/222

are called partial differentials. These partial differentials are sometimes denoted by d_xu, d_yu, d_zu, so that (56) is also written

$du=d_{x}u\ +\ d_{y}u\ +\ d_{z}u.$

Illustrative Example 1. Given $u=\arctan {\tfrac {y}{x}}$, find $dx$.

Solution.

${\frac {\partial u}{\partial x}}=-{\frac {y}{x^{2}+y^{2}}},{\frac {\partial u}{\partial y}}={\frac {x}{x^{2}+y^{2}}}.$

Substituting in (55),

$du={\frac {xdy-ydx}{x^{2}-y^{2}}}.$ Ans.

Illustrative Example 2. The base and altitude of a rectangle are 5 and 4 inches respectively. At a certain instant they are increasing continuously at the rate of 2 inches and 1 inch per second respectively. At what rate is the area of the rectangle increasing at that instant?

Solution. Let $x=$ base, $y=$ altitude; then $u=xy=$ area, ${\tfrac {\partial u}{\partial x}}=y,{\tfrac {\partial u}{\partial y}}=x.$

Substituting in (51),

(A)	${\frac {du}{dt}}=y{\frac {dx}{dt}}+x{\frac {dy}{dt}}.$

But $x=5$ in., $y=4$ in., ${\tfrac {dx}{dt}}=2$ in. per sec., ${\tfrac {dy}{dt}}=1$ in. per sec.

∴ ${\frac {du}{dt}}=(8+5)$ sq. in. per sec.$=13$ sq. in. per sec. Ans.

NOTE. Considering $du$ as an infinitesimal increment of area due to the infinitesimal increments $dx$ and $dy$, $du$ is evidently the sum of two thin strips added on to the two sides. For, in $du=ydx+xdy$ (multiplying (A) by $dt$),

$ydx$ = area of vertical strip, and

$xdy$ = area of horizontal strip.

But the total increment $\Delta u$ due to the increments $dx$ and $dy$ is evidently

$\Delta u=ydx\ +\ xdy\ +\ dxdy.$

Hence the small rectangle in the upper right-hand corner ($=dxdy$) is evidently the difference between $\Delta u$ and $du$. This figure illustrates the fact that the total increment and the total differential of a function of several variables are not in general equal.

127. Differentiation of implicit functions. The equation

(A)	$f(x,\ y)=0$

defines either $x$ or $y$ as an implicit function of the other.^[1] It represents any equation containing $x$ and $y$ when all its terms have been transposed to the first member. Let

(B)	$u=f(x,\ y);$
then	${\frac {du}{dx}}={\frac {\partial u}{\partial x}}+{\frac {\partial u}{\partial y}}{\frac {dy}{dx}}.$	(53), §125

1. We assume that a small change in the value of $x$ causes only a small change in the value of $y$.