SUCCESSIVE DIFFERENTIATION 101 



For example, to successively differentiate tan x, we want as 

 a general rule to be able to differentiate tan" x. 



y = tan n x = 2" where z = tan x 



-r- = nz n ~ l and ~ - sec 2 x 

 dz ax 



du 



-r- = nz n ~ l sec 2 x 



ax 



= n tan"- 1 x(\ + tan 2 x) 

 = n tan n-1 x + n tan n+1 x 



If we give n any integral value, we are enabled to differentiate 

 any integral power of tan x. 



When n = 1 y = tan x and -^ = 1 + tan 2 x 



flTF 



n = 2 ii = tan 2 x and -r- = 2 (tan a; + tan 3 #) 



//T* 



n = 3 v = tan 3 a- and -^ = 3 (tan 2 x + tan 4 x} 

 dec 



Successively differentiating tan x 

 y = tan x 



-r = 1 + tan 2 x 

 dx 



-j-| = 2 tan a? + 2 tan 3 a; 

 or 2 



-7-| = 2(1 + tan 2 a;) + 6 (tan 2 x + tan 4 x) 



CLXs 



= 2 + 8 tan 2 x + 6 tan 4 x 

 -T^J = 16 (tan x + tan 3 a?) + 24 (tan 3 x + tan 5 a?) 



= 16 tan x + 40 tan 3 x + 24 tan 5 x 

 -T^ = 16 (1 + tan 2 x} + 120 (tan 2 x + tan 4 a;) + 120 (tan 4 x + tan 6 a;) 



= 16 + 136 tan 2 x + 240 tan 4 x -r 120 tan 6 x 

 T-^ = 272 (tan x + tan 3 a?) + 960 (tan 3 a? +tan 5 a;) + 720 (tan 6 x 



uUv 



+ tan'a;) 



= 272 tan x + 1232 tan 3 x + 1680 tan 5 x + 720 tan 7 a; 

 Similarly, if we find the differential coefficients of cot n x, 

 sec x, tan n x, and cosec x, cot n #, we can use them to differentiate 

 successively cot x, sec x, and cosec x respectively. 



