(\partial)/(\partial y)(y/(sqrt(x^2+y^2)))

解

\frac{\partial}{\partial y} (\frac{y}{x ^{2} + y ^{2}})

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

解答ステップ

\frac{\partial}{\partial y} (\frac{y}{x ^{2} + y ^{2}})

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

\frac{\partial}{\partial y} (y) = 1

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

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

簡素化

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

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