44 DIFFERENTIAL COEFFICIENTS. 



Hence the rule for differentiating x n is universally esta- 

 blished. 



75. We shall now give some examples of the preceding 

 rules for finding differential coefficients. 



(1) Let y= smax. 



Put ax = z ; therefore y sin z, 



dy dy dz 



and -f =-f- --, Art. 63. 



ax dz ax 



But -f = cosz, Art. 51, 



dz 



and -, - = a, Art. 33, 



ax 



dy 



therefore -~ = a cos z = a cos ax. 



ax 



(2) Let y = sin (log x}. 



By logo; without any base specified, we mean log e x. 

 Put log x = z, 



therefore y = sin z, 



dy dy dz 



and -/-=-f -1-, Art. 63. 



ax 3 dx 



But -7^ = cos z, Art. 51, 



ds 



i = -, Art. 50, 

 dx x 



dy cosz cos (log a:) 

 therefore -f- = - = - ^ . 



dx x x 



(3) y= 



Put sin x = z, 



therefore y = log z, 

 dy dy dz 



1 cos a; 



= - cos x = = cot x. 

 z smx 



