Page 134 - Foundations Of Differential Calculus

6. On the Differentiation of Transcendental Functions 117
If p is any function of x, then in a similar way we have

d. sin p = dp cos p.

202. Similarly, if we are given cos x, that is, the cosine of the arc x, and we
are to investigate its differential, we let y = cos x and replace x by x + dx
so that y + dy = cos (x + dx). Since
cos (x + dx) = cos x · cos dx − sin x · sin dx,

and since, as we have just seen, cos dx = 1 and sin dx = dx,wehave

y + dy = cos x − dx sin x,
so that

dy = −dx sin x.
Hence, the differential of the cosine of any arc is equal to the negative of
the product of the differential of the arc and the sine of the same arc.
Hence, if p is any function of x, then

d. cos p = −dp sin p.

These differentiations can also be derived from previous results as follows.
If y = sin p, then p = arcsin y and

dy
.
dp = 2
1 − y
2

Since y = sin p, cos p = 1 − y , and we substitute this value to obtain
dp = dy/cos p, and so
dy = dp cos p
2

as before. In like manner, if y = cos p, then 1 − y = sin p and p =
arccos y. Hence
−dy −dy
= ,
dp = 2
1 − y sin p
so that we have as before

dy = −dp sin p.

203. If y = tan x, then
dy = tan (x + dx) − tan x;