[tex]y'=\frac{d}{dx}\left(-\frac{\sec x}{1+\tan x}\right)[/tex]
[tex]y'-\frac{d}{dx}\left(\frac{\sec x}{1+\tan x}\right)[/tex]
Aplicando a regra do quociente:
[tex]y'=-\frac{\frac{d}{dx}(\sec x)\cdot(1+\tan x)-\sec x\cdot\frac{d}{dx}(1+\tan x)}{(1+\tan x)^2}[/tex]
[tex]y'=-\frac{\sec x\tan x\cdot(1+\tan x)-\sec x\cdot\sec^2x}{(1+\tan x)^2}[/tex]
[tex]y'=-\frac{\sec x\tan x(1+\tan x)-\sec^3x}{(1+\tan x)^2}[/tex]
[tex]y'=\frac{\sec^3 x-\sec x\tan x(1+\tan x)}{(1+\tan x)^2}[/tex]
