42 DIFFERENTIAL COEFFICIENTS. 



72. Differential coefficient of z v . 

 Let y = z, where v and z are both functions of x. 

 Take the logarithms of both members of the equation, 

 hence 



log e y = vlog e z. 



Now since these two functions of x are always equal, their 

 differential coefficients with respect to x must be so. 



And = ^ 



ax ay dx 



= 1 ^, Art. 50. 

 y ax 



Also the differential coefficient of v log b z 

 dv , d log e z 



dv . v dz 



= s I ^ + ;s 



1 dv dv , ' v dz 



therefore y = Io S< 2 + i S> 



A dy_ (d?L i v _ &*\ 



dx " \das z dx) ' 



73. If we compare Arts. 29... 31 with Art. 72 we may 

 deduce a general rule for the differential coefficient of a 

 composite function. Differentiate in order each component 

 function, treating all the others as if they were constant; 

 then add the results thus obtained. 



It is advisable to call the attention of the student explicitly 

 to three different cases which beginners are apt to confound. 



(1) If y = z a where z is a function of x and a is a constant, 

 then by Arts. 47 and 63 



^ = as -i ffc e 



dx dx ' 



(2) Ify = a" where z is a function of x and a is a constant, 

 then by Arts. 49 and 63 



dy dz 



s -**? 



