120. La función $f\left( x \right)={}^{1}\!\!\diagup\!\!{}_{\left( {{x}^{2}}+1 \right)}\;$ tiene derivada

a)$f'\left( x \right)={}^{2}\!\!\diagup\!\!{}_{{{\left( {{x}^{2}}+1 \right)}^{2}}}\;$

b)$f'\left( x \right)={}^{2x}\!\!\diagup\!\!{}_{{{\left( {{x}^{2}}+1 \right)}^{2}}}\;$

c)$f'\left( x \right)={}^{-2x}\!\!\diagup\!\!{}_{{{\left( {{x}^{2}}+1 \right)}^{2}}}\;$

Calculemos dicha derivada como derivada de un cociente:

$f'\left( x \right)=\frac{\left( 1 \right)'\cdot \left( {{x}^{2}}+1 \right)-1\cdot \left( {{x}^{2}}+1 \right)'}{{{\left( {{x}^{2}}+1 \right)}^{2}}}=\frac{0\cdot \left( {{x}^{2}}+1 \right)-\left( 2x+0 \right)}{{{\left( {{x}^{2}}+1 \right)}^{2}}}=$

$=\frac{0-2x}{{{\left( {{x}^{2}}+1 \right)}^{2}}}=\frac{-2x}{{{\left( {{x}^{2}}+1 \right)}^{2}}}$