d/(dy)(y/(t^2+y^2))

解

\frac{d}{d y} (\frac{y}{t ^{2} + y ^{2}})

\frac{t ^{2} - y ^{2}}{( t ^{2} + y ^{2} ) ^{2}}

解答ステップ

\frac{d}{d y} (\frac{y}{t ^{2} + y ^{2}})

t

を定数として扱う

商の法則を適用:

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

= \frac{\frac{d y}{d y} ( t ^{2} + y ^{2} ) - \frac{d}{d y} ( t ^{2} + y ^{2} ) y}{( t ^{2} + y ^{2} ) ^{2}}

\frac{d y}{d y} = 1

\frac{d}{d y} (t^{2} + y^{2}) = 2 y

= \frac{1 \cdot ( t ^{2} + y ^{2} ) - 2 yy}{( t ^{2} + y ^{2} ) ^{2}}

1 \cdot (t^{2} + y^{2}) - 2 yy = t^{2} - y^{2}

= \frac{t ^{2} - y ^{2}}{( t ^{2} + y ^{2} ) ^{2}}