Calcolare la derivata delle seguenti funzioni:

1) $\displaystyle y= \frac {x^2-2}{e^x}$

2) $\displaystyle y= \frac {x}{x-2}$

3) $\displaystyle y=\frac{2}{x-3}$

4) $\displaystyle y=\frac{x^2+1}{x^2-4}$

SOLUZIONE

1) $\displaystyle y'(x)=\frac {2xe^x-(x^2-2)e^x}{e^{2x}}=\frac {2xe^x-e^xx^2+2e^x}{e^{2x}}=$

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

2) $\displaystyle y’= \frac {1\cdot (x-2)-x\cdot 1}{(x-2)^2}=\frac {x-2-x}{(x-2)^2}=-\frac {2}{(x-2)^2}$

3) $\displaystyle y’=\frac{0-2}{(x-3)^2}=-\frac{2}{(x-3)^2}$

4) $\displaystyle y’=\frac{2x(x^2-4)-2x(x^2+1)}{(x^2-4)^2}=\frac{2x^3-8x-2x^3-2x}{(x^2-4)^2}=$

$\displaystyle=-\frac{10x}{(x^2-4)^2}$