12 DIFFERENTIAL AND INTEGRAL CALCULUS. 



DIFFERENTIATION OF COMPLEX FUNCTIONS. 



The differentiation of complex functions can always be 

 made to depend upon that of simple functions by the 

 following method. 



Let y be a function of 2, z being a function of x, and 

 suppose that when x receives an increment A#, z receives 

 an increment As, and y in consequence an increment A# ; 



A?/ Ay Az , dy dy dz 



then since - - - - - always, -f- = -f- -. So also 

 Aa? A' Aa; ax dz ax 



dy dy du dx 



d ~ J 77" T ' a ^ n rough anv llum ber of steps 



that may be necessary before arriving at the simple func- 

 tion of x. These processes should, however, be performed 

 by a mental operation only 5 thus, if y log sin#, we may 

 put sin# = 2, and therefore y = logz, whence 



dy dy dz 1 cosic 



- = -r -y = - cos#= -s- =cotic. 



aa; dz dx z sin a? 



This should be written 



dy I , d . . 



-~ = -1 -=- (since) = cotie. 

 dx smx dx 



So 



= cot ( JTT + 4) sec 2 ({TT + J) (-Jw + Ja>) 



co 



sin 2 (ITT + Ja:) sin (-|TT + x) cos a? ' 



The differential coefficient of the sum of a number of dif- 

 ferent functions of x is obviously the sum of their differen- 

 tial coefficients ; that of a constant is zero, and that of a 

 function of x multiplied by a constant is the differential 



