Ex. 2.

..

dy = sec a tan x dx,

Du = a cos ax dx + b cos (b sec x) sec x tan x dx + dx.

df

dz

dx

Ex. 3.

u = xyz,

and z = sin (xy);

...

Du =

dr df

{(4) + (1) (1)} de + { (14) + (dz) (d) } dy.

Du = yz x12-1 dx + xyz log, x (y dz + z dy) ;

Dz (or dz) = cos (xy) (y dx + x dy);

dy

+ log ̧x xyz {z+xy cos (xy)} dy.

The cases of differentiation of functions of the preceding kind are so various and numerous, that it is impossible to discuss all; but the principles explained and illustrated as above are applicable universally, and from the examples given the student. ought to find no difficulty in solving other similar ones.

52.] In certain cases the subject variables enter under the functional symbols in particular combinations; and although PRICE, VOL. I.

M

the general principles unfolded above are applicable, yet the results take special forms; and certain relations exist between the several partial differentials of the given function by reason of the particular combination of the variables. Many cases of such functions will occur in the sequel, and we shall then have to consider them from a geometrical point of view: here I propose to exhibit their properties from an analytical point. And for this purpose some particular examples will serve best.

Let u be a function of two independent variables x and y; and suppose that x and y enter under the functional symbol in a certain combination, such as x divided by y: then

that is, u is a homogeneous function of x and y of O dimensions. Now whatever is the form of function which enters into the x-differential of u, the same function enters into the y-differential; although there will be different differential factors in each. Thus, for instance, if

u = sin

У

du

;

dy

replace

du

Ꮖ

y2

COS

by z, so that

y

du

u = f(z), and if

dz

so that, as we remarked above, the same form of function enters into both the partial derived functions; but the factors are different; and the relation between the partial differentials is, as deduced from (55),

and suppose f'(z) to be the derived function of ƒ (z); then

(56)

(57)

(58) being the partial derived functions; and (59) exhibiting

the relation between them.

Again, suppose the function to be

u = f(ax-by),

(60)

and suppose f'(z) to be the derived function of f(z); then from (60) we have

which are the partial differentials; and from them we have

which gives the relation between the partial derived functions. Or again, suppose the function to be of three variables; and let us assume it to be of the form

έ, and if the derived function of ƒ (§) is ƒ'(§), we have

= 2ƒ' (xy),

which are the three partial derived functions of (63).

The following are two other examples of the same kind.

53.] Thus far the functions which have been considered are explicit. Let us briefly consider one or two cases in which the variables are implicitly involved; and firstly the following, which is of importance in a future problem.

Ex. 1. Given z = r {y + x ƒ (z)}, where y and z are inde

dz

dz

pendent variables, it is required to find (d) and and

the relation between them.

dy

If y+xf(z) = &, let us symbolize the derived function of F() by F'; then we have

If

y-b

(64)

=, let us suppose the derived function of f() to be

f'() then from (64) we have

Or (64) might be put in the implicit form

u = F

(x-a, y-b).

C

(66)

Since z centers into both of the subjects of the function, it is

convenient for us at first to consider the z-c which is combined with y-b to be different to that which is combined with x — a; and let us therefore distinguish the second z by an accent, so that (66) becomes

u = F

X-a y
2-

(67)

and because that which is true when z is accented, that is, in the general case, will also be true in the particular case, that is, when the accent is omitted, the general result of (67) will include that of (66).

a and z

Now by reason of the mode in which c enter into the right-hand side of (67), it is evident that the x- and zpartial differentials of u will contain the same form of function, but that their differential factors will be different; let the form of function which enters into both, whatever it is, be represented x-a y; so that

by ф

In a similar way, the same form of function, whatever it is, will enter into the y- and the z-partial differentials of u, but the differential factors will be different in the two differentials; let x-a y

— be the form of function; then

Let us now consider the relation between these equations, when z. Suppose a functional equation to be given in the z

form

w = f (z, %);

(74)