What is the derivative of xtanx by using the first principle?

Dear Student,

y=xtanxLet y be the increment in y, corresponding to increase in x.y+y=x+xtanx+xSo,y+y-y=x+xtanx+x-xtanxy=xtanx+x+xtanx+x-xtanxy=xtanx+x-xtanx+xtanx+xyx=xtanx+x-xtanx+xtanx+xxOr, dydx=limx0yxdydx=limx0xtanx+x-xtanx+xtanx+xxdydx=limx0xtanx+x-xtanxx+limx0xtanx+xxdydx=limx0xsinx+xcosx+x-sinxcosxx+limx0tanx+xdydx=limx0xsinx+xcosx-sinxcosx+xcosx cosx+xx+tanxdydx=limx0xsinx+x-xcosx cosx+xx+tanxdydx=limx0sinxx limx0xcosx cosx+x+tanxdydx=1×xcosx cosx+tanxdydx=x sec2x+tanx
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