44.

To find the Differential Coefficient of cosec1 x.

y

=

cosec1x,

= X,

cosec y

y + dy = cosec1 (x + dx),

cosec (y+dy) = x + dx,

Ex = cosec (y + dy) – cosec y

Differentiation of Simple Functions of y with regard to x, y being a function of x.

45. Let u=y". Now, by Art. (19),

45'. We shall devote this section to the exemplification of the principles which have been established in Sections (1) and (2). The illustrations here given are not numerous: in order to acquire a practical familiarity with the processes of differentiation, as well as with the application of the general theorems which we shall develop in the subsequent pages of this work, it will be necessary for the student to have recourse to Peacock's or Gregory's Examples of the Differential Calculus.

=

Ex. 1. Let y sin a + sin ẞ + sin y, a, B, y, not involving x; then, the sum of the three sines being a constant quantity, we have, by Art. (11),

a being a constant quantity; then, by Arts. (12) and (30), we have

a and b being constants; then, by Arts. (13) and (31), we have

a being a constant; then, by Arts. (14), (30), and (32),

Ex. 7. Let y = x" . log, x. a*. sin 2. sin-1

x;

then, by Arts. (17), (30), (31), (32), (33), (39),

Ex. 9. If uy3, and y = cot x ; let it be proposed to find

du

dx