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LXXXV. Note on the Higher Derivative of a Function, the 

 variable of which is a Function of an independent variable. 

 By I. J. iSCHWATT *. 



IN E. R. Hedrick's translation of Goursat's work 

 'A Course in Mathematical Analysis ' appears the fol- 

 lowing problem (p. 32, 6.): — 



Show that the nth derivative of a function y = (p(u), where 

 tn's a function of the independent variable x, may be written 

 in the form 



<«) g =A lf („)+ £#»(,,) + + J^t-JKu), 



where 



,,s , d n u K k dV" 1 k(k-1) 2 d n u'- 2 . 



+ (-l)"-W^(«=l,2,....,n). 



[First notice that the ?2th derivative may be written in the 

 form (a), where the coefficients A 1? A 2 , ...., A n are inde- 

 pendent of the form of the function <j>(u). To find their 

 values, set <f>(u) equal to u, u 2 , . . . ., u n successively, and 

 solve the resulting equations for A ly A 2 , . . . ., A n . The 

 result is the form (/>)]. 



I have quoted the problem and the suggestions in full, and 

 shall now proceed to give several proofs for it, in the hope 

 that these proofs will illustrate certain operations with series 

 which might be useful in similar work. 



I. Let y = <j)(u), wherein u is a function of x. 

 Then 



dy _ dy du __,,,■ ^du 

 dx du ' dx ^ ^ ' dx 



dry ,., ,dhc ... N fdu\ 2 



d'fy ,,/ \d 3 u o,/// ^dud 2 u , , /;// ./du\ s 



a£ =* c«)ai+3*» a gj, +#'"M( a ). 



* Communicated by the Author. 



