(\partial)/(\partial x)(-(v(x,t))/(b(x,t)x-b(x,t)v(x,t)t))

解

\frac{\partial}{\partial x} (- \frac{v ( x , t )}{b ( x , t ) x - b ( x , t ) v ( x , t ) t})

- \frac{\frac{\partial}{\partial x} ( v ( x , t ) ) ( b ( x , t ) x - b ( x , t ) v ( x , t ) t ) - ( ( \frac{\partial}{\partial x} ( b ( x , t ) ) x + 1 \cdot b ( x , t ) ) - t ( \frac{\partial}{\partial x} ( b ( x , t ) ) v ( x , t ) + \frac{\partial}{\partial x} ( v ( x , t ) ) b ( x , t ) ) ) v ( x , t )}{( b ( x , t ) x - b ( x , t ) v ( x , t ) t ) ^{2}}

解答ステップ

\frac{\partial}{\partial x} (- \frac{v ( x , t )}{b ( x , t ) x - b ( x , t ) v ( x , t ) t})

t

を定数として扱う

定数を除去:

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

= - \frac{\partial}{\partial x} (\frac{v ( x , t )}{b ( x , t ) x - b ( x , t ) v ( x , t ) t})

商の法則を適用:

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

= - \frac{\frac{\partial}{\partial x} ( v ( x , t ) ) ( b ( x , t ) x - b ( x , t ) v ( x , t ) t ) - \frac{\partial}{\partial x} ( b ( x , t ) x - b ( x , t ) v ( x , t ) t ) v ( x , t )}{( b ( x , t ) x - b ( x , t ) v ( x , t ) t ) ^{2}}

\frac{\partial}{\partial x} (b (x, t) x - b (x, t) v (x, t) t) = (\frac{\partial}{\partial x} (b (x, t)) x + 1 \cdot b (x, t)) - t (\frac{\partial}{\partial x} (b (x, t)) v (x, t) + \frac{\partial}{\partial x} (v (x, t)) b (x, t))

= - \frac{\frac{\partial}{\partial x} ( v ( x , t ) ) ( b ( x , t ) x - b ( x , t ) v ( x , t ) t ) - ( ( \frac{\partial}{\partial x} ( b ( x , t ) ) x + 1 \cdot b ( x , t ) ) - t ( \frac{\partial}{\partial x} ( b ( x , t ) ) v ( x , t ) + \frac{\partial}{\partial x} ( v ( x , t ) ) b ( x , t ) ) ) v ( x , t )}{( b ( x , t ) x - b ( x , t ) v ( x , t ) t ) ^{2}}