154 DIFFERENTIAL COEFFICIENT 



be solved so as to give y in terms of x, say y = i/r (x}, then the 

 differential coefficient of y with respect to x can be found by 

 previous rules. If x can be expressed in terms of y, we can 



dx , ,, dy dx dy 



determine -y- and then -f- , since -j- x --- = 1. But as it is 

 dy dx ay dx 



often difficult, and sometimes impossible, to solve the given 

 equation, it is necessary to investigate a rule for finding -,- 

 which does not require this operation. 



Put u for < (x, y). From the given equation y is some 

 definite function of x ; hence 



fdu\ dy fdu 

 \dy) dx \dx, 



is, by Art. 172, the differential coefficient of u with respect to 

 x. But u is always zero, and therefore so also is its differential 

 coefficient with respect to x. Hence 



dy) dx \dxj ' 

 du\ 



, dy \dx) 



therefore -, = 5-rr 



dx /du\ 



178. This important result may also be obtained thus, 

 which is in effect combining into one Article portions of the 

 preceding pages. Let 



<t> & y) = 0. 

 Suppose x to become x -{ Ace and y to become y+Ay, so that 



< (x + Aic, y + Ay) = 0. 



Hence <f> (x + Ace, y + Ay) <f> (x, y] = 0, 



and </> (aj+Aa?, y+ Ay) -< (or+Az, y) +< (x+ Ace, y) -<f>(x, y)=0. 

 Divide by Ace, and we have 

 <}> (x+ Aa;, y+ Ay) -<ft (rg+ Aa;, y) Ay ^ ^(a?+Aa?,y)-^(a; > y) = a 



Ay Aa; Aa; 



This equation, being always true, remains so when Ao; and 

 Ay are diminished indefinitely. 



