Derivative rules

Possible derivation:
d/dx(tan^(-1)(sin(5 x)))
Using the chain rule, d/dx(tan^(-1)(sin(5 x))) = ( dtan^(-1)(u))/( du) ( du)/( dx), where u = sin(5 x) and ( d)/( du)(tan^(-1)(u)) = 1/(1 + u^2):
 = (d/dx(sin(5 x)))/(1 + sin^2(5 x))
Using the chain rule, d/dx(sin(5 x)) = ( dsin(u))/( du) ( du)/( dx), where u = 5 x and ( d)/( du)(sin(u)) = cos(u):
 = cos(5 x) d/dx(5 x)/(1 + sin(5 x)^2)
Factor out constants:
 = (5 d/dx(x) cos(5 x))/(1 + sin^2(5 x))
The derivative of x is 1:
Answer: | = (15 cos(5x)) / (1 + sin^2(5x))

Note  .......derivative of arctan (u)  =   1 / ( 1 + u^2) * du  .....so......

f(x)  =   arctan (sin (5x))   =   [  1 /  [ 1 + (sin(5x) )^2]  * 5 cos (5x)  =

[ 5 cos (5x) ]  /  [ 1 + sin²(5x) ]