(\partial)/(\partial t)(t/(c^2t^2-x^2))

解

\frac{\partial}{\partial t} (\frac{t}{c ^{2} t ^{2} - x ^{2}})

\frac{- c ^{2} t ^{2} - x ^{2}}{( c ^{2} t ^{2} - x ^{2} ) ^{2}}

解答ステップ

\frac{\partial}{\partial t} (\frac{t}{c ^{2} t ^{2} - x ^{2}})

c, x

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 t} ( t ) ( c ^{2} t ^{2} - x ^{2} ) - \frac{\partial}{\partial t} ( c ^{2} t ^{2} - x ^{2} ) t}{( c ^{2} t ^{2} - x ^{2} ) ^{2}}

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

\frac{\partial}{\partial t} (c^{2} t^{2} - x^{2}) = 2 c^{2} t

= \frac{1 \cdot ( c ^{2} t ^{2} - x ^{2} ) - 2 c ^{2} tt}{( c ^{2} t ^{2} - x ^{2} ) ^{2}}

1 \cdot (c^{2} t^{2} - x^{2}) - 2 c^{2} tt = - c^{2} t^{2} - x^{2}

= \frac{- c ^{2} t ^{2} - x ^{2}}{( c ^{2} t ^{2} - x ^{2} ) ^{2}}