d/(d{x)}(({x})/(sqrt({x)^2+{y)^2+{z}^2}})

解

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

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

解答ステップ

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

商の法則を適用:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

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

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

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

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

簡素化

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

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