(\partial)/(\partial x)(5/(x^2+y^2))

解

\frac{\partial}{\partial x} (\frac{5}{x ^{2} + y ^{2}})

- \frac{10 x}{( x ^{2} + y ^{2} ) ^{2}}

解答ステップ

\frac{\partial}{\partial x} (\frac{5}{x ^{2} + y ^{2}})

y

を定数として扱う

定数を除去:

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

= 5 \frac{\partial}{\partial x} (\frac{1}{x ^{2} + y ^{2}})

指数の規則を適用する:

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

= 5 \frac{\partial}{\partial x} ((x^{2} + y^{2})^{- 1})

連鎖法則を適用:

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

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

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

= 5 (- \frac{1}{( x ^{2} + y ^{2} ) ^{2}} \cdot 2 x)

簡素化

5 (- \frac{1}{( x ^{2} + y ^{2} ) ^{2}} \cdot 2 x) : - \frac{10 x}{( x ^{2} + y ^{2} ) ^{2}}

= - \frac{10 x}{( x ^{2} + y ^{2} ) ^{2}}