What is the derivative of xtanx by using the first principle - Maths - Limits and Derivatives

Question

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

Priyanka Kedia · Accepted Answer

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-xtanx△y=xtanx+△x+△xtanx+△x-xtanx⇒△y=xtanx+△x-xtanx+△xtanx+△x⇒△y△x=xtanx+△x-xtanx+△xtanx+△x△xOr, dydx=lim△x→0△y△x⇒dydx=lim△x→0xtanx+△x-xtanx+△xtanx+△x△x⇒dydx=lim△x→0xtanx+△x-xtanx△x+lim△x→0△xtanx+△x△x⇒dydx=lim△x→0xsinx+△xcosx+△x-sinxcosx△x+lim△x→0tanx+△x⇒dydx=lim△x→0xsinx+△xcosx-sinxcosx+△xcosx cosx+△x△x+tanx⇒dydx=lim△x→0xsinx+△x-xcosx cosx+△x△x+tanx⇒dydx=lim△x→0sin△x△x lim△x→0xcosx cosx+△x+tanx⇒dydx=1×xcosx cosx+tanx⇒dydx=x sec2x+tanx
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