Now, to proceed to the limit, putting n = an indefinitely large positive integer, and thereby rendering dx less than any assignable quantity, we have,

To find the Differential Coefficient of sin x with regard to x.

Now by the seventh Lemma of the first section of Newton's Principia we know that the arc and the chord of any curve vanish in a ratio of equality: whence it follows that the ratio between the sine and the circular measure of an angle is ultimately unity. Hence, in the limit,

To find the Differential Coefficient of sec x.

37. Putting y = sec x, we get

y + dy = sec (x + dx),

sec x

Sy = sec (x + dx)

To find the Differential Coefficient of cosec x.

38. Putting y = cosec x, y + dy = cosec (x + dx), we have

бу

= cosec (x + dx) – cosec x

To find the Differential Coefficient of sin1 x with regard to x.

Now, proceeding to the limit, when dx and dy assume values less than any assignable magnitudes, we have

To find the Differential Coefficient of tan1 x.

41. Proceeding in the same way as in the investigation of the differentials of sin1x and cos1 x, we have

Sx

y

-1