18 DIFFERENTIAL COEFFICIENT OF A PRODUCT. 



Now let h diminish without limit, aud we have 

 du _dy dz 

 dx dx dx' 



Hence the differential coefficient of the sum of two functions is 

 the sum of the differential coefficients of the functions. 



Similarly, if u = y z 



du _ dy dz 

 dx dx dx' 



28. The results of Art. 27 may be extended to the case 

 of any number of functions connected by the signs of addition 

 or subtraction. For example, let 



u w + y + z, 

 then, as before, AM = Aw 4- Ay -f Az ; 



AM Aw A;/ Az 

 therefore A^A^A^A^ 



therefore, proceeding to the limit, 



du _ dw dy dz 

 dx dx dx dx' 



29. Differential coefficient of the product of two Functions. 

 Let <f> (x} and ty (x) denote two functions of x, and let 



U = (j>(x) ^r(x). 



Change x into x + h, and let u + AM denote the new product, 

 then u + AM = < (x + h} ty (x + h), 



therefore AM = <f> (x + h} fy (x + h) <f> (x) ty (x) 

 = {<f>(x+h}-<j> (x}} ^(x + h}+<}> (x} 



,, AM $ 

 therefore -- = 



. 



Suppose now h diminished indefinitely; then the limit of 

 ^ is the differential coefficient of </>(#) with 



