CHAPTER V.

SUCCESSIVE DIFFERENTIATION.

77. In the preceding chapters we have shewn how from any given function of a variable another function may be deduced, called the differential coefficient of the first. This second function, by the same rules, has its differential coefficient, which is called the second differential coefficient of the original function.

Thus, if y=x", we have

coefficient of nx2-1 is n (n − 1) "-2, which is therefore the second differential coefficient of y or a". The second differential coefficient of y is denoted by

What we said of in Art. 26, we now say of

dy dx

that it is to be looked upon as a whole symbol, not admitting of decomposition into a numerator d'y and a denominator dx.

As day will be generally

will be generally a function of x it will have its dx2

differential coefficient. This is called the third differential coefficient of y, and is denoted by

d3y
da,

This process and notation may be carried on to any extent. The successive differential coefficients of a function are often conveniently denoted by accents on the function. Thus, if (x) be any function of x, then '(x), o" (x), 4"" (x), (x), &c. denote the first, second, third, fourth, &c. differential coefficients of (x) with respect to x.

IV

For

78. In some cases the nth differential coefficient of a function admits of a simple algebraical expression. example, suppose

where [n-1 stands for 1.2.3... (n − 1).

80. Differential coefficient of the product of two functions. Suppose

u = yz,

where y and z are functions of x; we have

So far, then, as we have proceeded, the numerical coefficients follow the same law as those of the Binomial theorem. We may prove by the method of induction that such will always be the case. For assume

+ n

+n

In+1, d"+1z dy drz dy drz d'y d"-1z
dxn+1 = y dxn+1 +
dx dx dx dx" dx dx-1+...
dr+1y d"+z\
dn―r+1z
[dx" dx"++1 + dx++1 dx"

+

+

n (n − 1) ... (n − r + 1) (d'y d2++1z

n (n − 1) ... (n − r) (dr+1y d2+z; dr+3y dn-r-1z)

+

Now the series (3) follows the same law as (1). Hence if for any value of n the formula in (1) is true, it is true also for the next greater value of n. But we have proved that it holds when n = = 3; therefore it holds when n=4, therefore when n=5, &c.; that is, it is universally true.

This theorem is called after the name of its discoverer, Leibnitz.

ах

81. If u = e cos bx; we have by Arts. 78 and 80,

We may also find another form for this nth differential coefficient as follows:

82. The following is an important example of Art. 80.