dy,

which corresponds to the ultimate value of the increment

fly', y) - fly, y.), and which we have represented by a distinct symbol Du, is evidently different from either of the numerators of the fractions du du dy,' dy, In order to obviate all possibility of confusion, we might use

du du the symbols d,u, dvd, to denote the numerators of

dy,' dy, the suffixes serving to point out the origin of the two differentials. Such a notation, however, although remarkably clear, would frequently be very embarrassing, especially in long operations. It will be sufficient for distinctness if we remember to regard du du and as fractions the denominators of which are insepa

dy, rably attached to the numerators, the symbols dy, and dy,, which express the denominators, thus serving to indicate the true nature of the du in the numerators. If, however, as will be sometimes convenient, we do put dy, u,

du du dynu, instead of du, in the expressions

we shall then

dy,' dy, be at liberty to treat these differential coefficients as ordinary algebraical fractions : thus Du_dy,u dy, dy,u dy,

(2), dx dy, "dx dy dx may be written, multiplying both sides of the equation by dx, Du = dy, u + dyzu

.. (3). The quantity dy,u denotes the differential of u taken with regard to y,, as if y, were constant, dy, the differential of u taken with regard to y, as if y, were constant, and Du the differential of u due to the simultaneous variations of y, and y, dependent upon

, the variation of x. The quantities dy, u, dugu, are called the partial differentials of u with regard to y., y,, respectively, and Du its total differential. The quantities

dy, u dy

2

+

U

2

dyu dyz

in the equation (2), or their equivalents

du du

dy,' dy,' in the equation (1), are called the partial differential coefficients

Du of u with regard to Yı, Yz, respectively. Finally, is called

dx the total differential coefficient of u.

The equation (3) shews that the total differential of u is equal to the sum of its partial differentials.

+

x

Differentiation of a Function of any number of Functions of a

single Variable. 21. Let u = f(91, , Yz), a function of three variables y., 42, 43, each of which is a function of x. Then if u', y', ya', ya, be

' simultaneous values of u, Yı, Y., Y;, we have

U = f (91, 92, Y.), u = f (yi, ya', y:'),
u' – u = f(yi', ya', y:) - f(y1, 42, 43),
u' – u f(y's yz, y) - f(y1, 92, Yz) y - y
x
yi - yi

x'
fly', ya', y) - f(y', yz, yz) Yı' – Ya

ya - y2
f(yi, ya', y:') - flyi', ya, y.) Y– :
Y3 – Y3

x' Proceeding to the limit we get, as in the case of two functions, Du

.....(1). dx

dyz . dx dyz dx

. du du

du In this equation

, are the partial differential dy,' dy,' dy

Du coefficients of u with regard to Yı, 92, Yz, respectively; and

dx the total differential coefficient of u with regard to x. adopt the suffix notation, the equation (1) may be written

Du _ d,,u dy duzu dy duzu dy,
du , ,

.(2)
da dy,
dx

dyz dx

+

1

du dy,

du dy,

du dyz

+

+

dy dx

Multiplying the equation (2) by dx, and putting

dynu

dyzu dy, = dy, u, dy, = dygu,

,

dy: = dyzu,

dugu dy,

=

=

dyz

dy

3

Yn

du dyn

+

+

dy.

yn

we have
Du = dy u + dyều + dou

..(3). The same theorem may evidently be extended to any number of functions; so that, if

u = f(y,, Y2, Yz, ...Yn) Y., Yu, Y. ..., being any n functions of x, then Du du dy, du dy, du dy,

dy. dx dy, dx dy,dx

dyz . dx

dy, dx or, replacing du in the expressions du du du

du dy,' dy,' dy, by d, u, d, u, d, u, d, u respectively, and clearing the equation of fractional forms,

Du = d u + d. u + d. u +
-

+ d u. It may therefore be stated as a general proposition, that the total differential of a function of any number of functions of a variable is equal to the sum of its partial differentials taken on the hypothesis of the separate variation of each of the several subordinate functions of the variable. Cor. If any one of the quantities y, Y, Y, .... Yn y for

YıYu Yz instance, be equal to x, which is the most simple form of functionality, then from the above demonstration it is plain that we may replace the corresponding term

du

du dy dx

by

dx thus if

U = f (x, y2, 43, 44, Du du du dy, du dys then dx da 'dy,' dx d. . dyz da

dx or, transforming the equation from differential coefficients to differentials,

Du = d u + dyzu + dyzu + .+ dy,U.

Differentiation of an implicit Function of a single Variable. 22. Let

V = 0 (x, y) = 0, y being therefore an implicit function of x: then c', x', y', being corresponding values of v, x, y, 0 = P(x, y) = 0, b' = + (x', y') = 0,

, 0 0
' o – v = $(', y') - (x, y) = 0,

or

+

X

=

o' °(2, y) – $(x, y) $(x, y) = y(x', Y)..=) = 0..(1). '- 0 ', '- '

( x - x

y - y Now in the limit, when a approaches indefinitely near to x, and therefore y' to y, we have

y' - y_dy

o'

Dv x - x dx' x - x

x

dx
$(x', y) - $(x, y) _d$(x,y)_dv
'

dx dx and, first replacing a' by x, and then y' by y,

φα', ') - (α)
$(x', y) - 0 (t', y) – $(x, y) - $(x, y) dø (x, y) dv

') (-
y - ญ

y' - Y

dy dy hence the equation (1) becomes

Do do
do dy

(2). dx dx dy ' dx This result gives us the expression for dy in terms of the

dx partial differential coefficients of v with regard to x and y taken

Y successively as separately varying. If we replace the symbol dv in the numerators of the fractions

dv dv

dx dy by the expressive forms dv, dv, we have, transforming the equation (2) from differential coefficients to differentials,

Do = d0 + d90 = 0. This result shews that if any function of x and y be always

+

= 0...

с

zero, its total differential or the sum of its partial differentials is always zero.

0

=

n

n

+

+

+

dy, dx

General Theory of the Differentiation of implicit Functions

of a single Variable. 23. If y, = 0, v, = 0, 0, = 0, ....0. = 0, v,

, where

.vm, are n functions of x, y,, Y2, 43,. .. •Yns the variables Y , Y , Yz,. .. .Ym, being therefore implicit functions

y.4243 of x, then

do do dy, do dy, do dys dv, dy,

dx dy . dx dyz . dx *dy, dx together with n - 1 additional equations involving «,, 03, 0, ... Um

V.Vn precisely as v, is involved in this.

Let x', yi, ya', ya'.. ...yn's o', be corresponding values of XY, Y X, Y, Y, Y; ...Y.,0,; then

Y3. YnV ų= f (x, ,, yz, Yz. ...Yn) = 0, ' = f(a', yi', ya', ya ....Y) = 0; hence 0 = v' – v, = f(x', yı', ya', ya'.. ...Yn')-f(x, 41, 42, 439. .yn)

= f (c', Y., Yz, Yz, ....Y) - f(x, Yı, Yz, Yz. ...Y) + f(x, y, Y 2 Y 3 .Y) - f(x', Yı, Ya, Y;,. .. .yn)

.42 + f(x, y, ya, y, . ...Yn) - f(x', y', Yz, Yz,. .. .Yn) '''

- , 8, 9,

1

2

=

+ f(x', yi', ya', ya......Y)-f(c', y', ya's Ya.......yn): whence, de, dyn, dyan. ...dem, denoting the partial increments of the functions to which they are prefixed with regard to X, Yı, Y ....Yn, respectively, and A the total increment of v,, we have

, , )
Av; _ : f (x, y2 Ya, Yg you.yn) + 8y, f (x", 91, 92, Yz;•••Yn) dy,

',
x

dy.

at dy, f(x', y', yz, Yz,...yn) dy,

Y dy,

8x