CHAPTER VIII.

SUCCESSIVE DIFFERENTIATION.

DIFFERENTIATION OF A

FUNCTION OF TWO VARIABLES.

124. WE have, in Art. 77, defined the second differential coefficient of a function to be the differential coefficient of the differential coefficient of that function. The differential coefficient of the second differential coefficient has been called the third differential coefficient, and so We are now

about to give another view of these successive differential coefficients.

125. Let

therefore

y = f(x),

y + Ay = f (x + h),

▲y = f (x + h) − ƒ (x).

In the right-hand member of the last equation change x into x+h and subtract the original value; we thus obtain

or

f(x+2h) − f(x + h) − { ƒ (x + h) — ƒ (x)},

-

ƒ (x + 2h) — 2ƒ (x + h) + f (x).

This result, agreeably to our previous notation, may be denoted by A(Ay), which we abbreviate into Ay. Hence ▲3y = f (x + 2h) — 2ƒf (x + h) +ƒ (x).

Similarly A (Ay) or A3y will be equal to

− {ƒ (x+2h) — 2ƒ (x + h) +ƒ (x)},

that is, A3y = f(x + 3h) → 3ƒ (x + 2h) + 3ƒ (x + h) − ƒ (x).

126. By pursuing the method of the last Article, we find expressions for Ay, Ay, &c. We shall not for our purpose require the general expression for A"y. It will, however, be easy for the reader to shew, by an inductive proof, that

▲"y=f(x+nh)—nƒ {x+(n−1) h} +

n(n-1)

1.2

......

-ƒ{x+(n−2) h} —...

± nf (x + h) = ƒ (x).

h3
3

h2

1.2

13

h3

ƒ'' (x) + — ƒ'' (x + 0 ̧h),

Hence

▲3y = h2f'"' (x) + " {4ƒ''' (x + 20h) — ƒ'''(x+ 0 ̧h)}.

Divide both sides by h2, that is (Ax)2, and then let h be diminished indefinitely. Hence we obtain

128. The result of the last Article may be generalized by the inductive method of proof. Assume

where

▲"y=h"f" (x)+h+1 f (x)

.(1),

(x) is a function of x and h, which remains finite when h is made = 0. From (1) we have

n+2

Ant1y = hn+1ƒnti (x) + h*+2 {} ƒn+2 (x + Oh) + f'(x + 0 ̧h)},

Equation (2) shews us that, granting the truth of (1), we can deduce for Antly a value of the same form as that we assumed for Ary. But Art. 127 gives for Ay an expression of the assumed form; hence A'y has the same form, and so also has Aty, and generally A"y.

From equation (1), by dividing both sides by h" and then diminishing hindefinitely, we have

129. Hitherto we have only considered functions of one independent variable; that is, we have supposed in the equation y=f(x), although quantities denoted by such symbols as a, b, &c. might occur in f(x), yet they were not susceptible of any change. Suppose now we have the equation

u = x2 + xy + y2,

and let y denote some constant quantity and x a variable, we have

du

dx

= 2x+y.

From the same equation, if x be a constant quantity and a variable, we obtain

y

Of course we cannot simultaneously consider x both constant and variable; but there will be no inconsistency if on one occasion and for one purpose we consider x constant, and on another occasion and for another purpose we consider it variable.

130. If x and y denote quantities such that either of them may be considered to change without affecting the other, they are called independent variables, and any quantity u, the value of which depends on the values of x and y, is called a "function of the independent variables x and y ;'

du

d'u d3u

3

dx' dx d3, &c., denote the successive differential coefficients of u, taken on the supposition that x alone varies; du d'u d3u denote the successive differential cody' dy' dys efficients of u, taken on the supposition that y alone varies.

y, then

du
dx

131. If u be a function of the independent variables x and will also be generally a function of x and y. Hence we may have occasion for its differential coefficient with respect to x or y. The former is denoted by

du dy dx

d2u

will be generally functions

of both x and y.

respect to x or y.

These may require to be differentiated with
Hence we use such symbols as

the meaning of which may be gathered from the preceding

remarks. For example,

d3u
dx dy dx

implies the performance

of three operations: we are to differentiate u with respect

to x, supposing y constant; the resulting function is to be differentiated with respect to y, supposing a constant: this last result is to be differentiated with respect to x, supposing y constant.

132. In considering the equation y = f(x), where we have one independent variable, the student could be referred to analytical geometry for illustrations of the nature of a dependent variable and of a differential coefficient. See Arts. 35-43. In like manner, if he is acquainted with the elements of geometry of three dimensions, he will be assisted in the present chapter of the Differential Calculus. For instance, the equation

z = ax+by+c

represents a plane; x and y are two independent variables, of which is a function. Here

&c., vanish.

...(1),

and all the higher differential coefficients,

2= √√ (x2 — x2 — y3).

Again, is the equation to a sphere. If we pass from a point on the sphere, whose co-ordinates are x and y, to another whose co-ordinates are x+Ax and y, we vary x without varying y. If in this case the value of the third co-ordinate be z + Az, we have

z+Az = √ {r2 — y2 — (x + ▲x)2} .

From (1) and (2) we can of course find

Az

(2).

A and its limit,

which we denote by dx'

dz

The process is the same as if we had given

where a is a constant; from which we deduce