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CHAPTER XL 



DIFFERENTIAL COEFFICIENT OF A FUNCTION OF FUNCTIONS 

 AND OF IMPLICIT FUNCTIONS. 



1G8. SUPPOSE u a function of y and z, and y and z them- 

 selves functions of x, it is required to find -y- . This of course 



might be obtained by substituting in u for y and z their values 

 in terms of ic, by which substitution u becomes an explicit 



function of x, and -j- can be found by previous methods. 



But it is often convenient to have a rule which gives -r- 



without requiring the substitution for y and z. To this rule 

 we proceed. 



169. Suppose u = <j> (y, z}, 



where y and z are both functions of x. Let x become x + Ax, 

 and in consequence let y, z, and u, become respectively y + Ay, 

 z + Az, and u + AM. Then 



therefore " = 



Aa; Ay Aa; 



<ft(y, z + kz}-(f>(i/,z) Az 

 &z ' Ax' 



Now let Ace and consequently Ay, Az, and AM, diminish 

 without limit ; then 



