14 DIFFERENTIAL AND INTEGRAL CALCULUS. 



from which the useless factor (a + x) J must be divided out ; 

 now, differentiating it as x$ (a + #)", we have 



Probably, however, it is better still to take the logarithm 

 of both sides before differentiating, as is done in proving 

 the rule, thus 



log?/ = f logo* - f log (a + a?) ; 



1 dy 1 f 3 5 ) 3a - 2z 

 therefore ~r = ~ v --- r = ?r r 

 y dx 2(x a 4 j 2aj (a + a?) ' 



% _ (3a - 2o;) aj* 

 <fo "' 2 (a + ic)J 



The last method certainly gives the least chance for 

 mistake. 



An expression of the form u v , where u and v are both 

 functions of a?, is best differentiated by taking its loga- 

 rithm. If 



e 1 dy dv , v du 



therefore ~T~ T log^-f - -^- 



y ax ax u ax 



dy . du dv 



or -f- = vu l -7- + u v locrw 7 . 



dx dx dx 



It is to be noticed that the two terms of this expression 

 are what we should have obtained by differentiating the 

 whole expression considering (1) v constant, (2) u constant, 

 and adding the results together j for if u were constant, 



the differential coefficient of u would be u v logu -j- , and 



if v were constant, vif 1 -7- . This is indeed an invariable 

 dx 



rule ; if y be a complex function of x, then, in whatever 

 way the various simple functions, of which it is composed, 

 be connected together, the complete differential coefficient 

 of y is the sum of all of what we may call partial diffe- 



