OF A FUNCTION OF FUNCTIONS. 151 



Here we cannot dispense with the brackets or some equi- 

 valent notation, f -*- J denoting what would be the differential 

 coefficient of u with respect to x, if y were not a function 



(l ?/ 



of x, and -v- denoting the actual differential coefficient of u 

 dx 



with respect to x, when y is a function of x. 

 173. For example, let u = tan' 1 (ccy), 



y = <?\ 



then (if\m y 



(dx- 



fdu 



dx 



du 



-j- - - 



ax 1 + x y 



therefore 



which of course is what we obtain if we differentiate tan" 1 (xe*) 

 with respect to x, 



174. Suppose u = <f>(v, y, z) where v, y, z, are each func- 

 tions of x. We have, as before, 



AM = (j> (v + Aw, y + Ay, z + A*) - (j> (v, y, z) 



, s + As) 6(v,y + Ay, z + As) 



-<f>(v,y,z + Aa) 

 (v, y, 2 + &z) - <f> (v, y, z} ; 

 AM _ <ft (v + At>, y + Ay, s + Az) - (p. y + Ay,z + As) Av 

 AJC ~ Av Aa; 



<j>(v,y + Ay, g + Aa) - <fr (P, y, g + Az] Ay 

 Ay Aa; 



<^> (v, y, z + Ag) - (, y, g) As 

 As Aa; ' 



