100 PRACTICAL MATHEMATICS 



the nth differential coefficient can be readily obtained, such as 

 x n , log e x, ef, a x , sin (ax + b), cos (ax + b), ef sin bx, &* cos bx. 

 To find the nth differential coefficient of ef sin bx. 



y = (P* sin bx 



= aef 3 * sin to 



= ef x (a sin bx + b cos bx) 



f777 



-r- = ad 3 * sin bx + be? cos bx 

 ax 



= Va 2 + b 2 ef sin (to + a), where tan a - - 



ci 



It appears in this example that differentiation is equivalent to 



multiplication by Va 2 + b 2 , and at the same time increasing the 

 angle by a. If this is so, then 



-^| = (a 2 + b 2 ) if* sin (bx + 2a) 

 dx 2 



-7-1 = (a 2 + 6 2 )^ e*" sin (bx + 3a) 



and = (a 2 + Z> 2 ) e"* sin (bx + na) 



d# w 



If this result is true for all integral values of n, it must be true 

 for (n + 1), and ^-| = (a 2 + 6 2 )^" e" sin {bx + (n+ l)a}, and 



d n ij 



this result can be established by differentiating -~ 



ax 



/Jn+ljj n 



and -r -z = (a 2 + b 2 ) 2 (ae sin (bx + na) + fce"* cos (to + na) } 



rt 



= (a 2 + b 2 ) 2 ef KK {a sin (bx + na) + 6 cos (to + na) } 

 = (a 2 + 6 2 )^ e * Va 2 + b 2 sin (to + na + a) 



n + l 



= (a 2 + b 2 ) 2 ft sin {to + (n + l)a} 

 which agrees with the anticipated result, and therefore 



~- = (a 2 + b 2 ) 2 e?* sin (to + na) where tan a = - 

 dx n a 



for all integral values of n. 



62. Other functions are of such a form that it is impossible to 

 obtain the nth differential coefficient, but the work of successive 

 differentiation can be made as simple as possible by working 

 to some general rule. Functions such as tan x, cot x, sec x, 

 cosec x, e* sec x, e? cosec x, can be treated in this way. 



