(\partial)/(\partial x)(1/((1+x^2+y^2)))

解

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

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

解答ステップ

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

y

を定数として扱う

指数の規則を適用する:

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

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

連鎖法則を適用:

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

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

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

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

簡素化

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

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