Question Number 102510 by Study last updated on 09/Jul/20

lim_(△x→0) ((sin^2 ((1/3)x+(1/3)△x)−sin^2 (1/3)x)/(△x))=?

Answered by bemath last updated on 09/Jul/20

lim_(Δx→0)  ((sin ^2 (1/3)(x+Δx)−sin ^2 (1/3)x)/(Δx)) =  (d/dx) [ sin ^2 ((1/3)x)] = (1/3)sin (((2x)/3))

Answered by Dwaipayan Shikari last updated on 09/Jul/20

Method 1  (2/3)sin(x/3)cos(x/3)=(1/3)sin((2x)/3)  Method 2  lim_(△x→0) (((sin((x/3)+△(x/3))+sin(x/3))(sin((x/3)+△(x/3))−sin(x/3)))/(△x))    lim_(△x→0) (((2sin(((((2x)/3)+△(x/3))/2))cos△(x/6))(2cos(((((2x)/3)+△(x/3))/2))sin△(x/6)))/(△x))  2sin((2x+△x)/3) .(((△x)/6)/(△x))=(1/3)sin((2x)/3)