Series. 19 l 



15, kxampi.ks. 



/. Find 5 successive derivatives of x\ 



2. Find 4 successive derivatives of .r^-f-.r^-f .I'-f ,v'-fa -f- 1. 



?. Find -x successive derivatives of — . 



\^x 



4. Find 3 successive derivatives of s/ i-{-x. 



5. Find 3 successive derivatives of yj i-^2x. 



6. Find 3 successive derivatives of s/ i-\-x-. 



7. Find T. successive derivatives of . 



I — X 



16, Theorem. Iji a function 0/ a binomial x-\-h, sarf(x-\-h), 

 ihe derivative with respect to .r, 2vhen h is regarded constajit, is eqiial 

 io the derivative with respect to h tvhen x is regai^ded consta7it. 



l^et v=/fx-{-hJ, and let x-\-h=z, then 



D, r=D,j'. D, 2. vSee XIII, Art. 25, Eq.fs) 

 But D,^=i. 



Hence D,j'=D.o'. (i) 



Again, D/,j'=D,j'. D/,r. See XIII, Art. 25, Hq.(5) 



But D/,2=i. 



Hence D/j'=D,j'. (2) 



From (i ) and (2) it follows that 



D,j/=D/,r. 



17, Taylor'vS Formula. We are now prepared to take up 

 Taylor's formula. 



If f(x-\-hJ can be developed into a series arranged according to 

 positive increasing powers of //, let us assume 



/,^.v+//;=A,.-f A/z + A/r-f A//'+ . . . (i) 



where A , A,, A,, etc., do not contain //, but are in general func- 

 tions of X. 



Take the derivative with respect to .r of each side of ( i ) and we 

 have 



D,/r.i- + //;==D,A -f//D,A, + //-D,A,-)-//a),A^+ . . . (2) 



Also take the derivative with respect to // of each side of (i) 

 and we have 



D/,/-^r+//; = A. + 2A//4-3A/r + 4A//^+ . . . (3) 



