Diffrentiate w.r.t x

Question

Differentiate:

\(\frac{d}{dx}cos^{-1}(\frac{x}{x+1})\)

Answer 1

\(\frac{d}{dx}cos^{-1}(\frac{x}{x+1})=\cfrac{-1}{\sqrt{1-(\frac{x}{x+1})^2}}\frac{d}{dx}(\frac{x}{x+1})\)

 = \(\frac{-(x+1)}{\sqrt{(x+1)^2-x^2}}\times\frac{(x+1).1-x(1)}{(x+1)^2}\) 

 = \(\frac{-(x+1)}{\sqrt{2x+1}}\times\frac1{(x+1)^2}\)

 = \(\frac{-1}{\sqrt{2x+1}(x+1)}\)