Differentiate the following functions with respect to `log((sinx)/(1+cosx))`

Explanation

Let `y=log (frac{sin x}{1+cos x})`


Differentiate it with respect to `x_{t}` we get


`frac{d y}{d x}=frac{d}{d x} log (frac{sin x}{1+cos x})`


`=frac{1}{(frac{sin x}{1+cos x})} times frac{d}{d x}(frac{sin x}{1+cos x}) ldots ldots . .[` Using chain rule ]


`=(frac{1+cos x}{sin x})[frac{(1+cos x) frac{d}{d x}(sin x)-sin x frac{d}{d x}(1+cos x)}{(1+cos x)^{2}}] ldots ldots` [Using quotient rule]


`=(frac{1+cos x}{sin x})[frac{(1+cos x)(cos x)-sin x(-sin x)}{(1+cos x)^{2}}]`


`=(frac{1+cos x}{sin x})[frac{cos x+cos ^{2} x+sin ^{2} x}{(1+cos x)^{2}}]`


`=(frac{1+cos x}{sin x})[frac{(1+cos x)}{(1+cos x)^{2}}]`


...