d/(dy)(y/(x^2+y^2+1))

解

\frac{d}{d y} (\frac{y}{x ^{2} + y ^{2} + 1})

\frac{- y ^{2} + x ^{2} + 1}{( x ^{2} + y ^{2} + 1 ) ^{2}}

解答ステップ

\frac{d}{d y} (\frac{y}{x ^{2} + y ^{2} + 1})

x

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} ( x ^{2} + y ^{2} + 1 ) - \frac{d}{d y} ( x ^{2} + y ^{2} + 1 ) y}{( x ^{2} + y ^{2} + 1 ) ^{2}}

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

\frac{d}{d y} (x^{2} + y^{2} + 1) = 2 y

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

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

= \frac{- y ^{2} + x ^{2} + 1}{( x ^{2} + y ^{2} + 1 ) ^{2}}