40 DIFFERENTIAL COEFFICIENTS 



1 J (x 2 



But sec y = x, therefore cos y = - , and sin y = 



OS 



x 

 see Art. 65, thus 



dx x *J (x 2 1) " 

 Similarly, if y = cosec" 1 ^, 



dy = 1 



/7/y /y . / ^^y ... 1 I 



iX-t-v M/ 'y V**' *- y 



69. In the manner given in the preceding Articles the 

 differential coefficients of the inverse trigonometrical functions 

 are usually determined. They may however be found without 

 using Art. 60. 



For example, suppose 



therefore y + &y = tan" 1 (x + h}, 

 therefore A# = tan" 1 (x + h) tan' 1 a; 



Ti 



,, f Ay 1. -! 



therefore -r- = - 7 tan 



Aa; A 



* 



tan J 



1 l+as(x + h) 



k 



1+x (x + h) 



Now let h diminish without limit, then 



* 



tan rr 



* t T -j. f I + x (x -\- n) .. 



the limit of -5 * 1 = l, Art. 21, 







1 + a; (a; + A) 



~J~ "^ "^ 2 



/T 1* I />** 



lX/t</ J. T^ </ 



