SPECIAL APPLICATIONS OF ANALYSIS 157 



7.351 Leibnitz's Theorem. If u and v are functions of x, 



d n (uv) d n v n du d n ~ l v n(n - i) d 2 u d n ~ 2 v 



dx n dX 



n(n - i) (n - 2) d*u d n ~ 3 v d n u 



3! " Jx* dx n ~ 3 + ...... +V dx^ 



7.352 Symbolically, 



d n (uv) , 



-j-J- = (u + v) W, 



where 



u = u, iP =v. 



7.353 ^ + 



dx n \ dxj 



7.354 If 4>[~j~} is a polynomial in , 



IH - ** ( + ) 



7.355 Euler's Theorem. If u is a homogeneous function of the nth degree of r 

 variables, xi, xz t . . . x r , 



/ d d d\ m 



(xi- -- \- x 2 - -- h . . . + %r -r I u = n m u, 



\ dxi dxz dx r / 



where m may be any integer, including o. 



7.36 Derivatives of Functions of Functions. 

 7.361 If /(*) = F(y), and y = </>(*), 



" / = if ^ W + J ^W + ^7 r<?J 



where 



d n , k d n . i y^(^ - i) 2 d" 



2 77 fr = -v fc -v M k 4- " 1r 



t/ * a^ w:y iKa*^ 7 2! y dx* 



7.362 



X \%/ ^ ' I ! 



(n - i) (n - 2) n(n-i) (n _ 2) /i\ 



~T~ ..In 2 ! I -v/ ' 



2. ( 



, n( - i) (n - 2) /<A- 

 + ( - i) (n - 2) ( - 3) - - 4- .... 



