38 DIFFERENTIAL COEFFICIENTS 



where the same symbols denote throughout the same quan- 

 tities. Hence, since 



Ay _ Ay &z 



A.-B ~~ Az Ax ' 

 we have 



-ty (z) <j)(x + Aa?) - <ft (a?) 



A./ A; A./' 



Now let Aa;, As, and Ay, diminish without limit, and we 

 obtain F' (x) = ty' (z) <' (x) ; 



or, as it may be written, 



dy __ dy dz 

 dx dz dx ' 



Hence the differential coefficient of y with respect to x is 

 equal to the product of the differential coefficient of y with 

 respect to z, and of the differential coefficient of z with respect 

 to x. 



64. "We may make a remark on the demonstration of the 

 last Article similar to that in Art. 61. It is often considered 



sufficient to say that " -^- = -~ x -r- by the properties of 

 fractions, and therefore, by taking the limit, ~ = -j- -r- ." 



65. Differential coefficient of shT'o;. 

 Let y = sin' 1 ^, therefore 



therefore -y- = cosy, Art. 51, 



therefore -3? = - , Art. 60. 



dx cos y 



Since siny = x, cosy=*J(lx*)', the proper sign to be 

 taken will of course depend on the value of y; we may there- 

 fore put 



dx VU-^)' 



remembering that the radical must have a negative sign if 

 cos y be negative. 



