Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/224

Let P be the point (x, y, z) on the surface given by the equation

(E)	$u=F(x,\ y,\ z)=0,$

and let PC and AP be sections made by planes through P parallel to the YOZ- and XOZ-planes respectively. Along the curve AP, y is constant; therefore, from (E), z is an implicit function of x alone, and we have, from (57a),

Point P on a surface.

(58)	${\frac {\partial z}{\partial x}}=-{\frac {\frac {\partial F}{\partial y}}{\frac {\partial F}{\partial z}}}.$

giving the slope at P of the curve AP, §122.

${\tfrac {\partial z}{\partial x}}$ is used instead of ${\tfrac {dz}{dx}}$ in the first member, since z was originally, from (E), an implicit function of x and y; but (58) is deduced on the hypothesis that y remains constant.

Similarly, the slope at P of the curve PC is

(59)	${\frac {\partial z}{\partial y}}=-{\frac {\frac {\partial F}{\partial y}}{\frac {\partial F}{\partial z}}}$

EXAMPLES

Find the total derivatives, using (51), (52), or (53), in the following six examples:

1. $u=z^{2}+y^{3}+zy,\ z=\sin x,y=e^{x}.$	Ans.	${\frac {du}{dx}}=3e^{3x}+e^{x}(\sin x+\cos x)+\sin 2x.$
2. $u=\arctan(xy),\ y=e^{x}.$	Ans.	${\frac {du}{dx}}={\frac {e^{x}(1+x)}{1+x^{2}e^{2x}}}.$
3. $u=\log(a^{2}-p^{2}),\ \rho =a\sin \theta .$		${\frac {du}{d\theta }}=-2\tan \theta .$
4. $u=v^{2}+vy,v=\log s,\ y=e^{s}.$		${\frac {du}{ds}}={\frac {2v+y}{s}}+ve^{s}.$
5. $u=\arcsin(r-s),r=3t,\ s=4t^{3}.$		${\frac {du}{dt}}={\frac {3}{\sqrt {1-t^{2}}}}.$
6. $u={\frac {e^{ax}(y-z}{a^{2}+1}}),\ y=a\sin x,z=\cos x.$		${\frac {du}{dx}}=e^{ax}\sin x.$
Using (55) or (56), find the total differentials in the next eight examples:
7. $u=by^{2}x\ +\ cx^{2}\ +\ gy^{3}\ +\ ex.$	Ans.	$du=(by^{2}\ +\ 2cx\ +\ e)dx\ +\ (2byx\ +\ 3gy^{2})dy.$
8. $u\ =\ \log x^{y}.$		$du={\frac {y}{x}}dx+\log xdy.$