d/(da)(a/(a^2+b^2))

解

\frac{d}{d a} (\frac{a}{a ^{2} + b ^{2}})

\frac{- a ^{2} + b ^{2}}{( a ^{2} + b ^{2} ) ^{2}}

解答ステップ

\frac{d}{d a} (\frac{a}{a ^{2} + b ^{2}})

商の法則を適用:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

= \frac{\frac{d a}{d a} ( a ^{2} + b ^{2} ) - \frac{d}{d a} ( a ^{2} + b ^{2} ) a}{( a ^{2} + b ^{2} ) ^{2}}

\frac{d a}{d a} = 1

\frac{d}{d a} (a^{2} + b^{2}) = 2 a

= \frac{1 \cdot ( a ^{2} + b ^{2} ) - 2 aa}{( a ^{2} + b ^{2} ) ^{2}}

1 \cdot (a^{2} + b^{2}) - 2 aa = - a^{2} + b^{2}

= \frac{- a ^{2} + b ^{2}}{( a ^{2} + b ^{2} ) ^{2}}