d/(dm)(1/2 mv^2)

解

\frac{d}{d m} (\frac{1}{2} m v (m)^{2})

\frac{v ^{2}}{2} + m v \frac{d v}{d m}

解答ステップ

\frac{d}{d m} (\frac{1}{2} m v^{2})

定数を除去:

(a \cdot f)^{'} = a \cdot f^{'}

= \frac{1}{2} \frac{d}{d m} (m v^{2})

積の法則を適用:

(f \cdot g)^{'} = f^{'} \cdot g + f \cdot g^{'}

f = m, g = v^{2}

= \frac{1}{2} (\frac{d m}{d m} v^{2} + \frac{d}{d m} (v^{2}) m)

\frac{d m}{d m} = 1

\frac{d}{d m} (v^{2}) = 2 v \frac{d v}{d m}

= \frac{1}{2} (1 \cdot v^{2} + 2 v \frac{d v}{d m} m)

簡素化

\frac{1}{2} (1 \cdot v^{2} + 2 v \frac{d v}{d m} m) : \frac{v ^{2}}{2} + m v \frac{d v}{d m}

= \frac{v ^{2}}{2} + m v \frac{d v}{d m}