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

解

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

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

解答ステップ

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

y, z

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

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

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

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

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

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