Find the derivative of (1-tan x/1+tan x)^1/2

Question

Find the derivation of. (1-tan x/1+tan x)^1/2

Answer 1

Let y=√((1-tan(x))/(1+tan(x))) = √((cos(x)-sin(x))/(cos(x)+sin(x))) = √((cos(x)-sin(x))^2/cos(2x)) (multiplying top and bottom by cos(x)-sin(x) and replacing cos^2(x)-sin^2(x) with cos(2x)).

y=(cos(x)-sin(x))/√(cos(2x)).

Let u=cos(x)-sin(x); du/dx=-(sin(x)+cos(x)).

Let v=(cos(2x))^-1/2;

dv/dx=-(1/2)cos(2x))^-3/2.(-2sin(2x))=tan(2x)/√cos(2x).

dy/dx=vdu/dx+udv/dx,

dy/dx=-(sin(x)+cos(x))/√cos(2x)+(cos(x)-sin(x))tan(2x)/√cos(2x).