Question

Найдите производную функцию
Y(x) =1/2sin^2(2x+п)​

Answer 1

$y(x)=\frac{1}{2}\cdot sin^2(2x+\pi )\\\\y'(x)=\frac{1}{2}\cdot 2\, sin(2x+\pi )\cdot (sin(2x+\pi ))'=sin(2x+\pi )\cdot cos(2x+\pi )\cdot 2=\\\\\star \; \; 2\cdot sina\cdot cosa=sin2a\; \; \star \\\\=sin(2\cdot (2x+\pi ))=sin(\underbrace {4x+2\pi }_{T=2\pi })=sin4x\\\\ili\\\\y(x)=\frac{1}{2}\cdot sin^2(2x+\pi )\\\\\star \; \; sin(\pi +a)=-sina\; \; ,\; \; (-sina)^2=(sina)^2=sin^2a\; \; \star \\\\y(x)=\frac{1}{2}\cdot sin^22x\\\\y'(x)=\frac{1}{2}\cdot 2\, sin2x\cdot cos2x\cdot 2=2\, sin2x\cdot cos2x=sin4x$