Page 143 - Foundations Of Differential Calculus

126 7. On the Differentiation of Functions of Two or More Variables
216. Now that we have found this general rule, by means of which func-
tions of howsoever many variables can be differentiated, it will be pleasant
to show its use in several examples.
I. If V = xy, then

dV = xdy + y dx.
x
II. If V = , then
y
dx xdy
dV = − .
y y 2
y
III. If V = √ , then
2
a − x 2
dy yx dx
dV = √ + .
2
2
2
a − x 2 (a − x ) 3/2
m n
IV. If V =(αx + βy + γ) (δx + y + ζ) , then
m−1 n
dV = m (αx + βy + γ) (δx + y + ζ) (αdx + βdy)
m n−1
+ n (αx + βy + γ) (δx + y + ζ) (δdx + dy) ,
or
m−1 n−1
dV =(αx + βy + γ) (δ + y + ζ)
by

(mαδ + nαδ) xdx +(mβδ + nα ) xdy +(mα + nβδ) ydx
+(mβ + nβ ) ydy +(mαζ + nγδ) dx +(mβζ + nγ ) dy.

V. If V = y ln x, then
ydx
dV = dy ln x + .
x
y
VI. If V = x , then
y−1 y
dV = yx dx + x dy ln x.
y
VII. If V = arctan , then
x
xdy − ydx
dV = .
2
x + y 2