(\partial)/(\partial t)(2/((x-3t)^2+1))

解

\frac{\partial}{\partial t} (\frac{2}{( x - 3 t ) ^{2} + 1})

\frac{12 ( - 3 t + x )}{( ( x - 3 t ) ^{2} + 1 ) ^{2}}

解答ステップ

\frac{\partial}{\partial t} (\frac{2}{( x - 3 t ) ^{2} + 1})

x

を定数として扱う

定数を除去:

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

= 2 \frac{\partial}{\partial t} (\frac{1}{( x - 3 t ) ^{2} + 1})

指数の規則を適用する:

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

= 2 \frac{\partial}{\partial t} (((x - 3 t)^{2} + 1)^{- 1})

連鎖法則を適用:

- \frac{1}{( ( x - 3 t ) ^{2} + 1 ) ^{2}} \frac{\partial}{\partial t} ((x - 3 t)^{2} + 1)

= - \frac{1}{( ( x - 3 t ) ^{2} + 1 ) ^{2}} \frac{\partial}{\partial t} ((x - 3 t)^{2} + 1)

\frac{\partial}{\partial t} ((x - 3 t)^{2} + 1) = - 6 (- 3 t + x)

= 2 - \frac{1}{( ( x - 3 t ) ^{2} + 1 ) ^{2}} (- 6 (- 3 t + x))

簡素化

2 - \frac{1}{( ( x - 3 t ) ^{2} + 1 ) ^{2}} (- 6 (- 3 t + x)) : \frac{12 ( - 3 t + x )}{( ( x - 3 t ) ^{2} + 1 ) ^{2}}

= \frac{12 ( - 3 t + x )}{( ( x - 3 t ) ^{2} + 1 ) ^{2}}