Differentiate the following with respect to `x :` `cos^(-1)(sinx)` and `cot^(-1)((1-x)/(1+x))`

Explanation

`text { Let } f(x)=cos ^{-1}(sin x)`

`Rightarrow f(x)=cos ^{-1}[cos (frac{pi}{2}-x)]`

`Rightarrow f(x)=frac{pi}{2}-x`

Thus, `f^{prime}(x)=frac{d}{d x}(frac{pi}{2}-x)=-1`

`cot ^{-1}(frac{1-x}{1+x})=frac{d}{d x}[tan ^{-1}(frac{1+x}{1-x})]`

`=frac{1}{1+(frac{1+x}{1-x})^{2}} times frac{1-x+1+x}{(1-x)^{2}}`

`=frac{(1-x)^{2}}{(1-x)^{2}+(1+x)^{2}} times frac{2}{(1-x)^{2}}`

`=frac{(1-x)^{2}}{1-2 x+x^{2}+1+2 x+x^{2}}`

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