Question

Умоляю решить найти производную функции в данном значении аргумента: y=4sin^5 2x ,x=п\8

Answer 1

$y = 4\sin^5(2x)\\\\\\ y' = (4\sin^5(2x))' = 4(\sin^5(2x))' =\\\\ 4*(2x)'*(\sin(2x))'*5*\sin^4(2x) = 40\cos(2x)\sin^4(2x)\\\\\\ y'(\frac{\pi}{8}) = 40\cos(\frac{\pi}{4})\sin^4(\frac{\pi}{4}) =\\\\ 40*\frac{\sqrt{2}}{2}*(\frac{\sqrt{2}}{2})^4 = 40*\frac{4\sqrt{2}}{32} = 40*\frac{\sqrt{2}}{8} = \boxed{ 5\sqrt{2} }$

Answer 2

у'=4·5·sin⁴2x·cos2x·2 = 40· sin⁴2x·cos2x = 40·sin⁴π/4·cosπ/4 = 40·¼·√2/2=5√2.