derivative of x/((x^2+y^2^{3/2)})

解

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

\frac{y ^{2} - 2 x ^{2} - 3 x \frac{d y}{d x} y}{( x ^{2} + y ^{2} ) ^{\frac{5}{2}}}

解答ステップ

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

商の法則を適用:

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

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

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

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

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

簡素化

\frac{1 \cdot ( x ^{2} + y ^{2} ) ^{\frac{3}{2}} - 3 ( x + y \frac{d y}{d x} ) ( x ^{2} + y ^{2} ) ^{\frac{1}{2}} x}{( ( x ^{2} + y ^{2} ) ^{\frac{3}{2}} ) ^{2}} : \frac{y ^{2} - 2 x ^{2} - 3 x \frac{d y}{d x} y}{( x ^{2} + y ^{2} ) ^{\frac{5}{2}}}

= \frac{y ^{2} - 2 x ^{2} - 3 x \frac{d y}{d x} y}{( x ^{2} + y ^{2} ) ^{\frac{5}{2}}}