DIFFERENTIAL COEFFICIENT OF A PRODUCT. 19 



respect to x, or <'(#); the limit of +-^ - j. ~ r ( x ) - g ^ 



n 



differential coefficient of ty (a;) with respect to x, or i/r' (x) ; 

 the limit of ty (x + h] is ty (x) ; 



therefore J = f (a?) ^ () + ^' (.) (). 



Hence /&e differential coefficient of the product of two functions 

 is found by multiplying each factor by the differential coefficient 

 of the other factor and adding the resulting products. 



Divide each side of the last result by u or < (x} ty (x) ; thus 



1 du _ 4>'(x) -^'(x}^ 

 udx <f> (x) i/r (x) ' 



30. An equation similar to that just obtained holds for the 

 product of any number of functions. For example, let 



u = wyz, 

 w, y, z being all functions of x. 



Assume v = wy, 



therefore u = vz; 



then, by Art. 29, 



1 du _ 1 dv \dz 

 u dx v dx. zdx' 



Proceeding in this manner we have as a rule : The differen- 

 tial coefficient of the product of any number of functions is 

 found by multiplying the differential coefficient of each factor 

 by all the other factors and adding the products thus formed. 



C2 



