OF A FUNCTION OF A FUNCTION. 37 



explicit function of x, and consequently y must have a dif- 

 ferential coefficient with respect to x. For example, if -z x* 

 and y = z 3 , we have by substitution y = x 6 . iNow this is 

 a function of x of which we know the differential coefficient, 



by Art. 44. Hence -f- = Qx s . But if z = cos # and y = a", we 



C13G '.,*- 



find y = a cosx , a function of x which we have not yet seen 

 how to differentiate. Hence the necessity and use of the rule 

 demonstrated in the next Article. - 



63. Differential coefficient of a function of a function. 



Let z = (f>(x) .. ................... ... (1), 



and y = ty(z) ........................ (2), 



so that y is a function of x ; required the differential coeffi- 

 cient of y with respect to x. 



Let x be changed into x + &x, iri consequence of which 

 z becomes z + Az, and suppose in consequence of this change 

 in 2, that y becomes y + Ay ; thus , 



(3), 

 (4). 



Now suppose that by putting for z its value in (2), we obtain 

 y = F(x] ....:,::;.. ................ (5), 



where F(x] denotes some function .of a;. From the mode 

 in which equation (5) is obtained it follows that we may 

 suppose x and y to have respectively the same values in (5) 

 as in (1) and (2), and also that 



y + ky = F(x + kx), .................. (6), 



where Aa; and Ay are the same quantities as have already 

 occurred in (3) and (4). 



From these equations we deduce 



Ay F(x + Ax) -Fix] 



=- - L 



from 



<f> (x + Ao?) d> (x} . .. ... 



= 21i - ^ r_i_Z from (1) and (3), 

 Aa; \ / \ /> 



