Find the derivative of tan (root x) from first principle - Maths - Limits and Derivatives

Question

Find the derivative of tan (root x) from first principle

Manbar Singh · Accepted Answer

Let y=tanxLet δy be an increment in y, corresponding to an increment δx in x
Then y+δy=tanx+δxδy=tanx+δx-y⇒δy=tanx+δx-tanx⇒δy=sinx+δxcosx+δx-sinxcosx⇒δy=sinx+δx×cosx-cosx+δx×sinxcosx+δxcosx⇒δy=sinx+δx-xcosx+δxcosx⇒δyδx=sinx+δx-xδx×1cosx+δxcosx⇒δyδx=sinx+δx-xx+δx-x×1cosx+δxcosx⇒δyδx=sinx+δx-xx+δx2-x2×1cosx+δxcosx⇒δyδx=sinx+δx-xx+δx-x×1x+δx+x×1cosx+δxcosx⇒limδx→0δyδx=limδx→0sinx+δx-xx+δx-x×1x+δx+x×1cosx+δxcosx⇒dydx=limδx→0sinx+δx-xx+δx-x×limδx→01x+δx+x×limδx→01cosx+δxcosx⇒dydx=1×1x+0+x×1cosx+0
 cosx⇒dydx=12x×1cos2x⇒dydx=sec2x2x

Find the derivative of tan (root x) from first principle.