Question

Найти производную второго порядка:

1) sin(x)/(1+sin(x))
2) xsin(x^3)
4) Частные производные второго порядка u=(x+y)/(x^2+y^2)

Answer 1

1) y(x) = sin x/(sin x + 1) = (sin x + 1 - 1)/(sin x + 1) = 1 - 1/(sin x + 1)
$y'= 0-frac{0-1*cos(x)}{(sin(x)+1)^2} =frac{cos(x)}{(sin(x)+1)^2}$

2) y = x*sin(x^3)
y' = 1*sin(x^3) + x*cos(x^3)*3x^2 = sin(x^3) + 3x^3*cos(x^3)

3) u = (x+y)/(x^2+y^2)
$frac{du}{dx} = frac{1(x^2+y^2) - (x+y)*2x}{(x^2+y^2)^2} = frac{-x^2-2xy+y^2}{(x^2+y^2)^2}$
$frac{du}{dy} = frac{1(x^2+y^2) - (x+y)*2y}{(x^2+y^2)^2}= frac{x^2-2xy-y^2}{(x^2+y^2)^2}$