Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/218

7. $u=x^{3}y^{2}-2xy^{4}+3x^{2}y^{3}$; show that $x{\tfrac {\partial u}{\partial x}}+y{\tfrac {\partial u}{\partial y}}=5u.$

8. $u={\tfrac {xy}{x+y}};$ show that $x{\tfrac {\partial u}{\partial x}}+y{\tfrac {\partial u}{\partial y}}=u.$

9. $u=(y-z)(z-x)(x-y)$; show that ${\tfrac {\partial u}{\partial x}}+{\tfrac {\partial u}{\partial y}}+{\tfrac {\partial u}{\partial z}}=0.$

10. $u=log(e^{x}+e^{y});$ show that ${\tfrac {\partial u}{\partial x}}+{\tfrac {\partial u}{\partial y}}=1.$

11. $u={\tfrac {e^{xy}}{e^{x}+e^{y}}};$ show that ${\tfrac {\partial u}{\partial x}}+{\tfrac {\partial u}{\partial y}}=(x+y-1)u.$

12. $u==x^{y}y^{x};$ show that $x{\tfrac {\partial u}{\partial x}}+y{\tfrac {\partial u}{\partial y}}=(x+y+\log u)u.$

13. $u=\log(x^{3}+y^{3}+z^{3}-3xyz);$ show that ${\tfrac {\partial u}{\partial x}}+{\tfrac {\partial u}{\partial y}}+{\tfrac {\partial u}{\partial z}}={\tfrac {3}{x+y+z}}.$

14. $u=e^{x}\sin y+e^{y}\sin x;$ show that

$\left({\tfrac {\partial u}{\partial x}}\right)^{2}+\left({\tfrac {\partial u}{\partial y}}\right)^{2}=e^{2x}+e^{2y}+2e^{x+y}\sin(x+y).$

15. $u=\log(\tan x+\tan y+\tan z);$ show that

$\sin 2x{\tfrac {\partial u}{\partial x}}+\sin 2y{\tfrac {\partial u}{\partial y}}+\sin {\tfrac {\partial u}{\partial z}}=2.$

16. Let $y$ be the altitude of a right circular cone and $x$ the radius of its base. Show (a) that if the base remains constant, the volume changes ${\tfrac {1}{3}}\pi x^{2}$ times as fast as the altitude; (b) that if the altitude remains constant, the volume changes ${\tfrac {2}{3}}\pi xy$ times as fast as the radius of the base.

17. A point moves on the elliptic paraboloid $z={\tfrac {x^{2}}{9}}+{\tfrac {y^{2}}{4}}$ and also in a plane parallel to the XOZ-plane. When $x=3$ft. and is increasing at the rate of 9 ft. per second, find (a) the time rate of change of $z$; (b) the magnitude of the velocity of the point; (c) the direction of its motion.

Ans. (a) $v_{z}=6$ ft. per sec.; (b) $v=3{\sqrt {13}}$ ft. per sec.; (c) $\tau =\arctan {\tfrac {2}{3}}$, the angle made with the XOY-plane.

18. If, on the surface of Ex. 17, the point moves in a plane parallel to the plane YOZ, find, when $y=2$ and increases at the rate of 5 ft. per sec., (a) the time rate of change of $z$; (b) the magnitude of the velocity of the point; (c) the direction of its motion.

Ans. (a) 5 ft. per sec.; (b) $5{\sqrt {2}}$ ft. per sec.; (c) $\tau ={\tfrac {\pi }{4}},$ the angle made with the plane XOY.

125. Total derivatives. We have already considered the differentiation of a function of one function of a single independent variable. Thus, if

$y=f(v)$ and $v=\phi (x),$

it was shown that

${\frac {dy}{dx}}={\frac {dy}{dv}}\cdot {\frac {dv}{dx}}.$