DIFFERENTIAL AND INTEGRAL CALCULUS. 13 



coefficient of the function, multiplied by the same con- 

 stant. 



(For if y = u l + u 2 + u a +... , Aj/ = Aw x + Aw 2 +... , 

 if # = a, A# = 0; 



and if y = aw, Ay = aAtt, 



J?/ du, du , efo 



and the values of -^ are -y-*- + -y- +... , zero, and a -j- 



dx dx ax ax 



respectively). The differential coefficient of the product of 

 any number of different functions of x is formed by differen- 

 tiating each function as if all the others were constant, and 

 adding the results ; for if 



I dy 1 JM, 1 du 9 1 c^w 



therefore -/ = - 7 + - ^ +...+ -- 7 - , 

 y ax u^ ax u z dx u n dx 



dy du. du du 



or = -35 " A - M " + " ^ v- " *;- 4 MA -"- 1& 



The rule for a quotient may be deduced from this r or may 

 be found by the same process, thus if 



u . \ dv \ du \ dv 



y = -, logy=tog-log, or----; 



, V -j -- U ~j- 



. - dy dx dx 



therefore - ., , 



dx v 



the result, which is often put into words as a rule to be 

 remembered, but, in my opinion, that process by no means 

 makes the rule easier to apply, and it is very doubtful if 

 it is worth while having a separate rule for a quotient. 

 Certainly if the index of the denominator be other than 

 unity, it is rather better to differentiate the whole function 



considering it a product. Thus if y = - - rr , we get, 



dy (a + x)-f fa-* - ouf 4 (a + a;)t 

 by the rule for quotients, -f- = 



' dx (a + x) 5 



