487. La función $f\left( x \right)=\frac{x}{{{x}^{2}}+1}$ tiene derivada

a) ${{f}^{'}}\left( x \right)=\frac{-{{x}^{2}}+1}{{{\left( {{x}^{2}}+1 \right)}^{2}}}$

b) ${{f}^{'}}\left( x \right)=\frac{1}{{{x}^{2}}+1}$

c) ${{f}^{'}}\left( x \right)=\frac{-2x}{{{\left( {{x}^{2}}+1 \right)}^{2}}}$

 

Calculando dicha derivada como derivada de un cociente tendremos:

 

$f'\left( x \right)\quad =\quad \frac{\left( x \right)'\cdot \left( {{x}^{2}}+1 \right)-\left( x \right)\cdot \left( {{x}^{2}}+1 \right)'}{{{\left( {{x}^{2}}+1 \right)}^{2}}}\quad =$

 

$=\quad \frac{1\cdot \left( {{x}^{2}}+1 \right)-\left( x \right)\left( 2x \right)}{{{\left( {{x}^{2}}+1 \right)}^{2}}}\quad =\quad \frac{{{x}^{2}}+1-2{{x}^{2}}}{{{\left( {{x}^{2}}+1 \right)}^{2}}}\quad =\quad \frac{-{{x}^{2}}+1}{{{\left( {{x}^{2}}+1 \right)}^{2}}}$