84 PRACTICAL MATHEMATICS 



We can use this result to establish other standard forms. 

 (1) If y = c* where c is a constant, 



y = <?* = ( e *y = c* 



e? = c or a = log e c 



where 

 Then 



-- 

 ax 



log e c 



(2) 



then 

 and 



x = e v 



= e v = x 



but 



dy 



dy _ 1 _ 1 



dx dx x 

 dy 



(3) If y = log a # where a is a constant, 



then 

 and 



but 



dx 

 dy 

 dy 



= a v log e a = x log e a 



dx dx x log e a 

 dy 



55. The Trigonometrical Functions. 

 (1) If y = sin (ax + b) where a and b are constants, 

 then y + By = sin{a(x + Bx) + b} 



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



( -, a Bx\ a &x 



= 2 cos ( ax + b H ) sin - 



V 2 / 2 



. a Sx 



s^ = a cos ( ax + b H ^ ) _ 



ox \ 2 / a 8x 



2 



. a Bx 

 sin 



Making Sa;infinitelv small. = = 1 



CL OCC 



and 



-r- = a cos 



o) 



