(\partial)/(\partial x)((3x+t)/(x-5t))

解

\frac{\partial}{\partial x} (\frac{3 x + t}{x - 5 t})

- \frac{16 t}{( x - 5 t ) ^{2}}

解答ステップ

\frac{\partial}{\partial x} (\frac{3 x + t}{x - 5 t})

t

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 x} ( 3 x + t ) ( x - 5 t ) - \frac{\partial}{\partial x} ( x - 5 t ) ( 3 x + t )}{( x - 5 t ) ^{2}}

\frac{\partial}{\partial x} (3 x + t) = 3

\frac{\partial}{\partial x} (x - 5 t) = 1

= \frac{3 ( x - 5 t ) - 1 \cdot ( 3 x + t )}{( x - 5 t ) ^{2}}

簡素化

\frac{3 ( x - 5 t ) - 1 \cdot ( 3 x + t )}{( x - 5 t ) ^{2}} : - \frac{16 t}{( x - 5 t ) ^{2}}

= - \frac{16 t}{( x - 5 t ) ^{2}}