Question

Derivative of (x sinx + cosx)/(x sinx - cosx) wrt x.

Answer 1

\(\frac{d}{dx}\frac{xsin x+cosx}{xsinx-cosx}\) = \(\frac{(xsin x-cosx)d/dx (xsinx+cosx)-(xsin x+cosx)d/dx (xsinx-cosx)}{(xsinx-cosx)^2}\) 

\(=\frac{(xsinx-cosx)(xcosx + sinx-sinx)-(xsinx + cosx)(xcosx+sin x+sinx)}{(ssinx-cosx)^2}\)

\(=\frac{x^2sinx cosx-xcos^2x-(x^2sinx cosx + 2xsin^2x+xcos^2x+2sinx cosx)}{(xsonx-cos x)^2}\) 

\(=\frac{-2xcos^2x+2xsin^2x+2sin xcosx}{(xsinx - cosx)^2}\)\(=\frac{-2x(cos^2x-sin^2x)+sin2x}{(xsin x-cosx)^2}\) 

\(=\frac{sin2x-2xcos2x}{(xsinx - cos x)^2}\)