d/(dq)((q^1q^2)/(r^2))

解

\frac{d}{d q} (\frac{q ^{1} q ^{2}}{r ( q ) ^{2}})

\frac{q ^{2} ( - 2 q \frac{d r}{d q} + 3 r )}{r ^{3}}

解答ステップ

\frac{d}{d q} (\frac{q ^{1} q ^{2}}{r ^{2}})

商の法則を適用:

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

= \frac{\frac{d}{d q} ( q ^{1} q ^{2} ) r ^{2} - \frac{d}{d q} ( r ^{2} ) q ^{1} q ^{2}}{( r ^{2} ) ^{2}}

\frac{d}{d q} (q^{1} q^{2}) = 3 q^{2}

\frac{d}{d q} (r^{2}) = 2 r \frac{d r}{d q}

= \frac{3 q ^{2} r ^{2} - 2 r \frac{d r}{d q} q ^{1} q ^{2}}{( r ^{2} ) ^{2}}

簡素化

\frac{3 q ^{2} r ^{2} - 2 r \frac{d r}{d q} q ^{1} q ^{2}}{( r ^{2} ) ^{2}} : \frac{q ^{2} ( - 2 q \frac{d r}{d q} + 3 r )}{r ^{3}}

= \frac{q ^{2} ( - 2 q \frac{d r}{d q} + 3 r )}{r ^{3}}