SECTION 6.-Successive differentiation of implicit functions.

84.] Firstly, let us consider a function of two variables x and y, and of the form,

u = F(x, y) = c,

The latter may be deduced from (121) by differentiating with respect to the equicrescent variable x, as well as from (120), whence we have deduced it.

d3y

Similarly from D3u=0 may be deduced ; and from sub

dx3

sequent total differentials the other derived-functions of y may

be formed.

dy

85.] Ex. 1. Given y2-2yx+a2 = u = 0, to find and

day

dx dx2

Instead however of introducing formally the general values of dy d2y

and given in equations (121) and (122), it is more conda dx2 venient to differentiate the given function immediately according to the principles contained in Art. 48, and illustrated in the last three examples of that Article; the two following examples illustrate the process.

d2y
dx2

(2x+ady) (y3 + ax) − (2 y dy + a) (x2+ay)

(y3+ax)

dx

Ex. 3. If x2+ y2=a2; to find

and

dx

dx2

86.] Hence also, if an implicit function is given involving x and y, we may calculate the several coefficients of the powers of x in Maclaurin's Theorem, see Art. 57, and thus expand y in a series of terms of ascending powers of x.

Let the given implicit function be u= F(x, y) = c; then, since in the series (8), Art. 57, f(0), ƒ'(0), ƒ"(0), ... are seve

the notation signifying that particular values of the coefficients are taken, viz. when x = 0. In applications however of this theorem it is much less laborious to calculate the successive values directly from the given equation than to substitute in the preceding formulæ.

d2y

dx2

За

(3 ay3 — x) + dy (6 ay dy − 1) — dy = 0, ƒ”(0) = 0;

dx

dx

day (6ay dx
dy
1-1)

dx2

+

dx

•'. y=1+

2 x3

За 1 27 a3 1.2.3

which is one of the values of y in terms of a deduced from the

given cubic equation; the other two values may be found by taking the other two impossible cube roots of unity in the value of ƒ (0).

and the three series give three different values of y, which are the three roots of the given cubic equation in terms of x.

An equation also may often be put in other algebraic forms, and then expanded by Maclaurin's Theorem in a different form of developement, and sometimes in a series of descending powers of x. As, for instance, consider the last example,

Expanding which, as in the last two examples, and taking only the possible cube root, we have

and replacing y and x by their values, and multiplying through by at, we have

1

1

87.] The above theory is also applicable to the expansion of a function which is of great importance in the calculation of many series; and gives some numerical coefficients known by the name of Bernoulli's Numbers.

and as every odd power of x, which entered into the expansion of ƒ (x), would also enter into that of ƒ (x) —ƒ(-x), it follows that no odd power of x enters into f(x) except the first.

the last term being the nth derived-function of (125), and being

calculated by the method of Art. 55. Let x = 0 in these several equations, and we have

f(0) = f(0), which is an identity;

5ƒ'(0)+10ƒ'""'(0)+10ƒ”(0) +5ƒ′(0) +ƒ(0) = 0, ƒ''(0) =

1

30'

ƒn−2 (0) + ...... + nƒ′(0) +ƒ(0) = 0. (126) fn-2 (0)+

The last is the general equation, by means of which ƒn-1 (0) may

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