(\partial)/(\partial x)(x/(x^7-y^4))

解

\frac{\partial}{\partial x} (\frac{x}{x ^{7} - y ^{4}})

\frac{- 6 x ^{7} - y ^{4}}{( x ^{7} - y ^{4} ) ^{2}}

解答ステップ

\frac{\partial}{\partial x} (\frac{x}{x ^{7} - y ^{4}})

y

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} ( x ) ( x ^{7} - y ^{4} ) - \frac{\partial}{\partial x} ( x ^{7} - y ^{4} ) x}{( x ^{7} - y ^{4} ) ^{2}}

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

\frac{\partial}{\partial x} (x^{7} - y^{4}) = 7 x^{6}

= \frac{1 \cdot ( x ^{7} - y ^{4} ) - 7 x ^{6} x}{( x ^{7} - y ^{4} ) ^{2}}

1 \cdot (x^{7} - y^{4}) - 7 x^{6} x = - 6 x^{7} - y^{4}

= \frac{- 6 x ^{7} - y ^{4}}{( x ^{7} - y ^{4} ) ^{2}}