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

解

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

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

解答ステップ

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

a, t

を定数として扱う

定数を除去:

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

= t \frac{\partial}{\partial x} (\frac{1}{a ^{2} t ^{2} - x ^{2}})

指数の規則を適用する:

\frac{1}{a} = a^{- 1}

= t \frac{\partial}{\partial x} ((a^{2} t^{2} - x^{2})^{- 1})

連鎖法則を適用:

- \frac{1}{( a ^{2} t ^{2} - x ^{2} ) ^{2}} \frac{\partial}{\partial x} (a^{2} t^{2} - x^{2})

= - \frac{1}{( a ^{2} t ^{2} - x ^{2} ) ^{2}} \frac{\partial}{\partial x} (a^{2} t^{2} - x^{2})

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

= t (- \frac{1}{( a ^{2} t ^{2} - x ^{2} ) ^{2}} (- 2 x))

簡素化

t (- \frac{1}{( a ^{2} t ^{2} - x ^{2} ) ^{2}} (- 2 x)) : \frac{2 t x}{( a ^{2} t ^{2} - x ^{2} ) ^{2}}

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