Question

differentiate xsin x + cosx/sin x - xcos x with respect to x from first principle.

Answer 1

y =(xsinx+cosx)/(sinx-xcosx)

differentiate w.r.t x
dy/dx={(sinx-xcosx).d (xsinx+cosx)/dx-(xsinx+cosx).d (sinx-xcosx)/dx}/(sinx-xcosx)^2

={(sinx-xcosx)(sinx+xcosx-sinx)- (xsinx+cosx)(cosx-cosx+xsinx)}/(sinx-xcosx)^2

={(sinx-xcosx)(xcosx)-(xsinx+cosx)(xsinx)}/(sinx-xcosx)^2

={xsinxcosx-x^2cos^2x-x^2sin^2x-xsinxcosx}/(sinx-xcosx)^2

=-x^2 (sin^2x+cos^2x)/(sinx-xcosx)^2

= -x^2/(sinx-xcosx)^2