[Risolto] Calcolo derivata

1.

$3x^2+4x+4$

2. Derivata del quoziente

$\frac{(x^2+1)-x(2x)}{(x^2+1)^2}=\frac{1-x^2}{(x^2+1)^2}$

 3. Derivata del prodotto

$2x(x^2-1)+x^2(2x)=4x^3-2x$

Ciao!

$$ y = x^3+2x^2+4x $$

$y' = 3 x^{3-1}+2\cdot 2 x^{2-1} + 4 x^{1-1} = $

$= 3x^2+4x+4 $

$$ y = \frac{x}{x^2+1} $$

$y' = \frac{ D(x) (x^2+1) - x D(x^2+1)}{(x^2+1)^2} = \frac{ 1 \cdot (x^2+1)- x (2x)}{(x^2+1)^2} = $

$= \frac{x^2+1-2x^2}{(x^2+1)^2} = \frac{-x^2+1}{(x^2+1)^2}$

$$ y =x^2 (x^2-1) $$

$ y ' = D(x^2)(x^2-1) + x^2 D(x^2-1) = $

$= 2x (x^2-1)+x^2(2x) = 2x^3-2x+2x^3 = -2x $