Derivativks. 179 



It is easy to see that this result is correct, for in the equation 

 1'=^-', substitute the value of r and we have 



j'=^r^ + 2/=j»^4-4-^+4- 

 Hence D,v=^x^-\-Hx. 



26. Examples. 



Find the derivative with respect to x of the following ex- 

 pressions : 



7. (x^^ax-\-d)\ 



3. (x-\-ar^2(x-^a). 



4. (2X-\-i)'-\-^(2X-^2,)-\r\. 



6. 2(x'-i)'-\-^(x'-ir + (x^-i). 



27. To FIND THE Derivative with Respect to x of any 

 Positive Integral Power of x. 



Let y=x\ ' (i) 



Give to X the value x-\-Jx and we get 



y-\-Jy^(x-^^x)\ (2) 



Expanding the right-hand member of (2) we get 



y+Jy=-x-'-^nx"-'Jx-^'-^^^~^Kx"-'(Jxy-\- . . . -¥(^^x)-. (t,) 



Subtract (i) from (t^) and we get 



Jy=^nx'-JxV'^''~^^x''-^(Jx)^-i- . . . +(^Kv)\ (a,) 



Divide both sides of ( /^) by -'.r and we get 



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



Y)^y=.nx"-\ (6) 



Reasoning exactly as above we could show that when y^=^nx", 



D,y^nax"~\ 

 This fonnula maj- be expressed in words thus — 

 The denvathc ivHh respect to x of ax" is found by multiplying 

 the expo?ie7if by the eoefficierJ and reduci^ig the expone7it \ . 



