I need help differentiating the following. I am not sure if what I have done is correct.

1. $\displaystyle \frac{d}{dx}\frac{1}{1+\sin(\cos x)} $

$\displaystyle =\frac{1}{[1 + \sin(\cos x)]^2}[\cos(\cos x)\sin x] $

2. $\displaystyle \frac{d}{dx} \log (1 + \sin(1 + \tan x))$

$\displaystyle = \frac{1}{1 + \sin(1 + \tan x)}\cos(1 + \tan x)\sec^2 x $

Could someone tell me if what I have done is correct or not.