Question Number 104350 by bemath last updated on 21/Jul/20

lim_(△x→0)  ((sin ((α+△x)^n )−sin (α^n ))/(cos ((α+△x)^n )sin (α+△x)−cos (α^n )sin (α)))

Answered by john santu last updated on 21/Jul/20

lim_(△x→0)  ((sin ((α+△x)^n )−sin (α^n ))/(△x))  . lim_(△x→0)  ((△x)/(cos ((α+△x)^n sin (α+△x)−cos (α^n )sin (α)))=  (1) f(α)=sin α^n  ⇒f ′(α)=nα^(n−1) cos (α^n )  (2) g(α)=cos (α^n )sin (α)  → g ′(α)=−nα^(n−1) sin (α^n )sin(α)+  cos (α^n )cos (α)  Therefore the result is   ((nα^(n−1) cos (α^n ))/(−nα^(n−1) sin (α^n )sin (α)+cos (α^n )cos (α)))  (JS ⊛)