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

解

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

- \frac{x y}{( x ^{2} + y ^{2} ) ^{\frac{3}{2}}}

解答ステップ

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

x

を定数として扱う

定数を除去:

(a \cdot f)^{'} = a \cdot f^{'}

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

指数の規則を適用する:

\frac{1}{a} = a^{- 1}

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

連鎖法則を適用:

- \frac{1}{2 ( x ^{2} + y ^{2} ) ^{\frac{3}{2}}} \frac{\partial}{\partial y} (x^{2} + y^{2})

= - \frac{1}{2 ( x ^{2} + y ^{2} ) ^{\frac{3}{2}}} \frac{\partial}{\partial y} (x^{2} + y^{2})

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

= x (- \frac{1}{2 ( x ^{2} + y ^{2} ) ^{\frac{3}{2}}} \cdot 2 y)

簡素化

x (- \frac{1}{2 ( x ^{2} + y ^{2} ) ^{\frac{3}{2}}} \cdot 2 y) : - \frac{x y}{( x ^{2} + y ^{2} ) ^{\frac{3}{2}}}

= - \frac{x y}{( x ^{2} + y ^{2} ) ^{\frac{3}{2}}}