Derivatives. 173 



Expanding, 



y-{-Jy=^x^+SxJx-h4f'^-^-r-^5- (3) 



Subtract (i) from (t,) and we obtain 



Jy=-SxJx+4(Jx)\ (4) 



Divide both sides of (4.) by Jx and we get 



J=8.v+4J-r. (5) 



Taking the limit of each side as Jx approaches zero we get 



or B,y=8x. (7) 



We notice that the result is exactly the same as equation (S) 



in the first example under Art. 10, except that x appears here 



where a appeared before. 



We will hereafter proceed as we have just done and will usually 



find D.r y as a function of x, but occasionally, as in Art. 5, D^y 



will turn out to be a constant. 



(6. Derivative of a Constant. 



Let j'=a constant, then as x does not appear in the expression 



for r, X maybe changed at pleasure and the change does not affect 



r, or Jx may have any value whatever, but Jy is always zero. 



Jy 

 Hence -^=0, 



^x 



therefore D.v,r=o. 



17. To FIND THE Derivative with Resect to x of the 

 Algebraic Sum op two Functions of x. 



Let one function of x be represented by u and the other by v, 

 and let their sum be represented by )'; then 



r=w+7'. (1) 



When X is increased by Jx suppose that n is increased by Jw, 

 V is increased by Jz' and y is increased by J)', then after x is 

 increased by Jx we have 



y^Jy=U-{-Ju-{-Z'-hJv. (2) 



Subtract (i) from (2) and we get 



Jy^Ju^Jv. (7,) 



