Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/226

25. A covered water tank is made of sheet iron in the form of an inverted cone of altitude 8 ft. surmounted by a cylinder of altitude 5 ft. The diameter is 6 ft. If the sun's heat is increasing the diameter at the rate of .002 ft. per min., the altitude of the cylinder at the rate of .003 ft. per min., and the altitude of the cone at the rate of .0025 ft. per minute, at what rate is (a) the volume increasing; (b) the total area increasing?

In the remaining examples find ${\frac {dy}{dx}}$, using formula (57a):

26. $(x^{2}+y^{2})^{2}-a^{2}(x^{2}-y^{2})\ =\ 0.$	Ans.	${\frac {dy}{dx}}=-{\frac {y}{y}}\cdot {\frac {2(x^{2}+y^{2})-a^{2}}{2(x^{2}+y^{2})+a^{2}}}.$
27. $e^{y}-e^{x}+xy\ =\ 0.$		${\frac {dy}{dx}}={\frac {e^{x}-y}{e^{x}+x}}.$
28. $\sin(xy)-e^{xy}-x^{2}y\ =\ 0.$		${\frac {dy}{dx}}={\frac {y[\cos(xy)-e^{xy}-2x]}{x[x+e^{xy}-\cos {xy}]}}.$

128. Successive partial derivatives. Consider the function

$u=f(x,\ y);$

then, in general,

${\frac {\partial u}{\partial x}}$ and ${\frac {\partial u}{\partial y}}$

are functions of both $x$ and $y$, and may be differentiated again with respect to either independent variable, giving successive partial derivatives. Regarding $x$ alone as varying, we denote the results by

${\frac {\partial ^{2}u}{\partial x^{2}}},{\frac {\partial ^{3}u}{\partial x^{3}}},{\frac {\partial ^{4}u}{\partial x^{4}}},\cdots ,{\frac {\partial ^{n}u}{\partial x^{n}}},$

or, when $y$ alone varies,

${\frac {\partial ^{2}u}{\partial y^{2}}},{\frac {\partial ^{3}u}{\partial y^{3}}},{\frac {\partial ^{4}u}{\partial y^{4}}},\cdots ,{\frac {\partial ^{n}u}{\partial y^{n}}},$

the notation being similar to that employed for functions of a single variable.

If we differentiate $u$ with respect to $x$, regarding $y$ as constant, and then this result with respect to $y$, regarding $x$ as constant, we obtain

${\frac {\partial }{\partial y}}\left({\frac {\partial u}{\partial x}}\right),$ which we denote by ${\frac {\partial ^{2}u}{\partial y\partial x}}.$

Similarly, if we differentiate twice with respect to $x$ and then once with respect to $y$, the result is denoted by the symbol

${\frac {\partial ^{3}u}{\partial y\partial x^{2}}}.$