150 DIFFERENTIAL COEFFICIENT 



made in finding the values of these differential coefficients. 

 Hence the above equation should be written 



du _ fdu\ dy fdu\ dz 

 dx \dyt dx \dz ) dx ' 



Of course the brackets may be omitted, and indeed frequently 

 are omitted, provided we can feel certain of remembering the 

 conditions which they are designed to express. The beginner 

 will do well to use them, although as he advances in the 

 subject he may be able to dispense with them. 



171. For example, let u = z z + y 3 + zy, 



z = sin x 



then 



dx~ ' 



dz 



-j- = cos x ; 

 ax 



therefore -y- = (3# 2 + z) & + (2z + y) cos x 



= (%e** + sin x) e* + (2 sin x + e*} cos x 

 = Be 9 " + e* (sin x + cos x) + sin 2x ; 



and this value is of course precisely what we obtain if we 

 substitute in u for y and z their values in terms of x, thus 

 obtaining u = e 331 + e* sin x + sin 2 x, and then differentiate with 

 respect to x. 



172. An important case of the general proposition is 

 obtained by supposing z = x so that -y- = 1. We have then 



du _ (du\ dy fdu\ 

 dx ~ \dy) dx \dxj ' 



