Page 137 - Foundations Of Differential Calculus

120 6. On the Differentiation of Transcendental Functions
have as follows:
y = sin x, z = cos x,
dy = dx cos x, dz = −dx sin x,
2 2 2 2
d y = −dx sin x, d z = −dx cos x,
3 3 3 3
d y = −dx cos x, d z = dx sin x,
4 4 4 4
d y = −dx sin x, d z = dx cos x,
....

206. In a similar way we can find the differentials of all orders of the
tangent of the arc x. Let y = tan x = sin x/cos x and keep dx constant.
Then
sin x
y = ,
cos x
dy 1
= ,
2
dx cos x
2
d y = 2 sin x ,
3
dx 2 cos x
3
d y = 6 4 ,
2
4
dx 3 cos x − cos x
4
d y = 24 sin x − 8 sin x ,
3
5
dx 4 cos x cos x
5
d y = 120 120 + 16 ,
4
6
2
dx 5 cos x − cos x cos x
6
d y = 720 sin x 480 sin x + 32 sin x ,
5
7
3
dx 6 cos x − cos x cos x
7
d y 5040 6720 2016 64
8
2
4
6
dx 7 = cos x − cos x + cos x − cos x .
207. Any function whatsoever in which the sine or cosine of an arc is
involved can be differentiated by these rules. This can be seen from the
following examples.
I. If y = 2 sin x cos x = sin 2x, then
2
2
dy =2dx cos x − 2dx sin x =2dx cos 2x.

1 − cos x 1
II. If y = ,or y = sin x, then
2
2
dx sin x
.
dy =
2 2(1 − cos x)