DIFFERENTIATION 85 



(2) If y = cos (ax + b) where a and b are constants, 

 then y + By = cos{a(# + Bx) + b} 



By = cos (ax + b+ a Bx) cos (ax + b) 



f a Bx\ a Bx 



= - 2 sin lax + b+ ^-J sin 



. a Bx 



js 5 sm - 



o v . / , a ox \ 2 



/ a x \ ~ ' 



= a sin ( ax + b 4 ) 



\ 2 / a 



Bx 



Making Bx infinitely small, *- 



CL OX 



du 



and - = a sin (ax + b) 



dx 



(3) If y = tan (ax + b) where a and & are constants, 



then 



sin (ax + b) 



11 = - - - - 



cos (ax + b) 



and 



Bx} + b} 



~ _ 

 y 



cos{a(x+&x) + b} 



sin (ax + b+ a Bx) sin (ax + b) 



cos (ax + b+ a Bx) cos (oo; + b) 

 sm(ax + b + aBx) cos(ax + b)cos (ax + b + a Bx) sin(ax+b) 



cos (ax + b + a Bx) cos (ax + b) 

 sin a Bx 



cos (ax + b+ a Bx) cos (o# + b) 



sin a # 



By a Bx 



Bx cos (ax + b+ a Bx) cos (a# + b) 



Making Bx infinitely small ' K- = 1 



du a 



and -2 = . = a sec 2 (ax + b) 



dx cos 2 (ax + b) 



These results enable us to work with the trigonometrical func- 

 tions of all forms of angles, by properly adjusting the constants 

 a and /;. 



If 6 = the angle becomes ax, the ordinary multiple angle. 



If b = and a = 1, the angle is simply x. 



