178 AlvGlSBRA. 



25. To FIND THE Derivative with Respect to x of a 

 P\tnction of Another Function of x. 



Suppose y is some function of 2, and 3- is some function of .r, 

 then ultimately y is a function of x, hence it has a derivative 

 with respect to x. 



But as y is directly a function of z it has a derivative with re- 

 spect to Z'. 



Moreover, as s is a function of x it has a derivative with respect 

 to -V. 



We have identically 



'Jx Jz Jx' * -^ 



Taking the limit of each side as Jx approaches zero, remem- 

 bering that the limit of the product of two variables equals the 

 product of their limits, we have 



limit j Jy \ _ limit \ Jy ] limit \ ■^^' ( . . 



^•^- ^ o ) Jx \ ~Jx :: o ) j^ \ ' Jx:: o I "j;;f c ' ^>' 



Now z being a function of x we may write 



z=/(x). 

 and if x be increased by Jx we have 



z-\-Jz=/(x-^Jx), 

 and from this it is evident that, as Jx approaches zero, Jz must 

 also approach zero. Hence 



limit { ^V\ _ limit ^ Jy] 



Jx^ o|/^i - J,o o( j-;^) <3; 



Substitute from (t,) in (2) and we have 



limit \ Jy\ _ limit f Jy ) limit ( -J^ ] 



•J-^" :^ oiJx\ '~^'^ :■ o I j^ I • Jx : o I j^ \ '4; 



The left-hand member of (4.) is D,j; the first factor of the 

 right-hand member is D.j, for it is just the same as the left-hand 

 member except that z everywhere takes the place of x; and the 

 second factor of the right-hand member is D, 2-. 



Hence B^y=D,y . B,3. (s) 



If y=^;- and 2=x'^-\-2, then 



D„.j/=2.2 and D,z=2x. 

 Hence b}^ equation (^) 



D, J' = 2.2- . 2X= ^zx= ^x(x^ -f 2 j = 4^- ' -f 8x. 



